Seven-sided star figures and tuning algorithms in Mesopotamian, Greek, and Islamic texts

Texts Source: Archiv für Orientforschung , 2011, Bd. 52 (2011), pp. 121-155 Published by: Archiv für Orientforschung (AfO)/Institut für Orientalistik Stable URL: https://www.jstor.org/stable/24595107 JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at https://about.jstor.org/terms is collaborating with JSTOR to digitize, preserve and extend access to Archiv für Orientforschung This content downloaded from 192.87.31.155 on Sun, 28 Jan 2024 08:56:42 +00:00 All use subject to https://about.jstor.org/terms

Seven-Sided Star Figures and Tuning Algorithms in Mesopotamian, Greek, and Islamic Texts By Joran Friberg! (Gothenburg) 1. Regular Polygons and Star Figures in Greek and Mesopotamian Texts In Euclid’s Elements, propositions XIII.7-12 deal with the following types of regular n-sided polygons with n = 3, 5, 6, 10: the equilateral triangle, the pentagon, the hexagon, and the decagon. (See, most recently, the discussion in Friberg, Amazing Traces, Sec. 7.2.) In Hero of Alexandria’s Metrica I, the sections 17-25 are devoted to rules for the (approximate) computation of the areas of regular n-sided polygons when n = 5, 6, 7, 8, 9, 10, 11, 12. (See Heath, HGM II, 326-329.) Finally, according to Lucian and a scholiast to the Clouds of Aristophanes, “the triple interwoven triangle, the pentagram, i.e. the star-pentagon, was used by the Pythagoreans as a symbol of recognition between the members of the same school, and was called by them Health” (Heath, HGM I, 161). This means that also the n-sided regular star figure with n = 5, the pentagram, was known (and probably studied) by early Greek mathematicians. One purpose of the present paper is to give a brief berg, MSCT 1), namely: n = 3: examples 1, 2, 3; n survey of all known instances when n-sided regular = 4: examples 3 and 5; n = 6: example 2. Two come polygons or star figures occur, in one form or another, from the Iraq Museum in Baghdad, courtesy F. Alon Mesopotamian clay tablets from the Ist, 2nd, and Rawi, namely both examples with n = 8. Examples 2- 3rd millennia BC. The results of the survey are pre- 3 with n = 7 appear, explicitly and implicitly, in a text sented in tabular form in Fig. 1.1 below. The tabular discussed quite recently by Horowitz in JANES 30 and survey shows n-sided regular polygons with n = 3, 4, by Waerzeggers and Siebes in NABU 2007/2. 5, 6, 7, and n-sided star n=4 n=3 n = 5, 7, 8, figures with n=6 n=5 In the tabular survey, the > Pas |Fi following notations are $ used regarding the type c EDIII: t OB: d of the texts: this is the most likely OB: c < à. OB: c, d n=7 n=8 As ZA Y4 x y OB” d Sel: d OB, LB: p form of a figure mentioned in a mathematid this is the most likely intended form of a figp ure shown in a geometric diagram, OB: c, d Kass: p this is the most likely cal problem text, y this is the most likely OB: d form of a figure mentioned in a mathematit À \V cal table of constants, OB: c, d, p DL JO» pr-Sum: d Neo-Sum?: d OB: d OB: d form of a figure related to entries in a mathematical table text. Note that a substan. tial a part of the examples Pai P listed in the tabular sur- > LB: d, p texts in the Scheyen collection (Fri- È N 2 EDII:d OB, NB: d, p vey come from recently published 7 7 N 7 7 BG xs ') This work has been supported by Stiftelsen Langmanska Kulturfonden. OB: d OB: Fig. 1.1. Regular polygons and star figures appearing on known Mesopotamian clay tablets. Archiv für Orientforschung 52 (2011)

The more detailed discussion below of the various examples listed in the tabular survey in Fig. 1.1 will start with the examples with the smallest number of of the two equilateral triangles had the given lengths 1 (- 60) and 10, respectively. The following curiously formulated entry in the sides, proceeding from the several known cases with n Old Babylonian table of constants = 3 to the single known case with n = 12. a rule for the computation of the area of an equilateral G = IM 52916 gives triangle: A peg-head (triangle), with an eighth torn out, 26 15 its 2.n = 3: Equilateral Triangles constant G rev. 7' What this means is that for an equilateral triangle with Equilateral triangles are three-sided regular polygons. An equilateral triangle inscribed in a circle appears on the Old Babylonian clay tablet MS 3051 (Friberg, MSCT 1, Fig. 8.1.1; see Fig. 2.1 below). It is likely that the school boy who drew the diagram on the tablet had been given an assignment to compute the areas of the four parts of the divided circle, the equilateral triangle and the three circle segments, when the length of the circumference was given, equal to precisely 1 (- 60 length units). the side s the area A is A = s/2 - (sqs. 3)/2 - s = (appr.) s/2 - (1 - 1/8) - s = ;30 + (1 - 307 30) : sq. s = 326 15 - sq. s. (See Friberg, MSCT 1, Sec. 8.2). This rule is explicitly applied in the Kassite (post-Old-Babylonian) mathematical text MS 3876 (Friberg, MSCT 1, Sec. 11.3), as one of the steps in the correct computation of the weight of the shell of a colossal icosahedron, composed of (6 - 1) : 4 = 20 equilateral triangles of copper, each one of them with the side 3 cubits and the Indeed, each one of the three circle segments would then be bounded by a circular arc of length 60/3 = 20, as correctly indicated in the diagram. According to a convenient Babylonian convention, the length of the diameter of the circle would be 60/3 = 20, as well, and the length of the radius r = 10. Consequently, the area thickness 1 finger (= 1/30 cubit); see Fig. 2.3. On the obverse of the Old Babylonian tablet TMS 2 (Fig. 5.1 below), the area of an equilateral triangle with the side 30 (one sixth of the area of a regular hexagon with this side) is given as 6 33 45, which is correctly 1/4 of 26 15, the mentioned ‘constant’ for an of the equilateral triangle would be A = s/2 : h, where h = r + r/2 = s 15 and = r - sqs. 3 10 - sqs. 3.2 (See Friberg, MSCT 8.2.4.) However, the 1, Fig. incorrect value recorded inside the equilateral triangle in the diagram is 1 52 30, carelessly computed as A=h/2-h=15/2- 15 =7;30- 15 = 1 52;30. With departure from this incorrect result, the area of each one of the three circular segments is then computed as area of segment = (area of circle Fig. 2.1. MS 3051. An equilateral triangle inscribed in a circle. - area of triangle)/3 = (5 00 - 1 52;30)/3 = 3 07;30/3 = 1 02;30. obv. This incorrect value is recorded inside each one of the three circular segments. An equilateral triangle divided into a chain of three trapezoids plus a smaller equilateral triangle is depicted on MS 2192 (Friberg, MSCT 1, Fig. 8.2.2; see Fig. 2.2 below). In this case, the assignment probably was to find the areas of the various parts of the larger triangle when the sides 2) The abbreviation sqs. stands for “squareside” or, in modern terms, “square root.” Fig. 2.2. MS 2192. An equilateral triangle divided into three trapezoids and a smaller equilateral triangle.

sqs. 3 = (appr.) 2 : (1 - 1/8) = 7/4 and sqs. 3 = (appr.) 2 - (1 - 1/10 1/30) = 26/15. See the thorough discussion of more or less accurate Greek and Babylonian square side approximations in Friberg, Amazing Traces, Ch. 16.) A geometric doodle on the reverse of an Old Babylonian tablet with a single multiplication table on the obverse has the form of a badly drawn “upside-down” equilateral triangle divided into several smaller pieces by two lines parallel to the top and at least two diagonal Fig. 2.3. MS 38762. A colossal icosahedron made lines. See Fig. 1.1, bottom (Friberg, MSCT 1, of (6 - 1) - 4 = 20 equilateral copper triangles. Fig. 8.1.14). equilateral triangle, meaning the area of an equilateral triangle with the side 1 (- 60). In the Late-Babylonian mathematical “recombination text” W 23291 (Friberg, BaM 28, 285-286), two rules are given for the computation of the area of an equilateral triangle. In § 4 b, the rule is formulated in the following way: 1 peg-head-field, equilateral, that with an 8th torn out, stroke steps of ditto and steps of 26 15 go. 3. n = 4: Squares Squares are four-sided regular polygons. The oldest known appearance of squares on any clay tablet from Mesopotamia can be found on a tablet from Suruppak, dateable to the Early Dynastic Illa period (c. 2600-2500 BC). The tablet, VAT 12593, is inscribed with a metro-mathematical table of areas of This is clearly a reformulation of the Old Babylonian large squares, with side lengths expressed as multiples rule mentioned above. (Here, ‘peg-head’ means ‘trianof the ninda (Friberg, MSCT 1, Fig. 6.1.3). gle,’ and ‘stroke steps of ditto and steps of 26 15 go’ Also from the Early Dynastic period, but somewhat means ‘multiply the side length by itself and by 26 younger than VAT 12593, and of unknown prove- 15.°) The rule is followed by a diagram and a numerinance, is CUNES 50-08-001 (Friberg, MSCT 1, 419- cal application of the rule; see Fig. 2.4. 425, Figs. A7.1-2). It is a very large and complex metro-mathematical table of areas of squares, divided into a series of sub-tables with the side lengths of the he 1 £e 1 3 Squares expressed as multiples of the ninda and various fractions of the ninda. o a M ON un WwW 1 front 1 sag Fig. 2.4. W 23291 § 4 b. An equi- Younger still, from the Early Dynastic IIIb period, is the smaller, but parallel, text A 681 from Adab (Friberg, MSCT 1, 357-60, Fig. A1.4), a table of areas of squares with side lengths expressed as multiples of lateral triangle with the side the cubit. 1 (- 60) and the height 52;30 Some = (1 - 1/8) - 60. Interestingly, the rule in § 4 b, obviously a legacy from Old Babylonian mathematics, is confronted in § 4 c with a more accurate, presumably Late-Babylonian rule: 1 peg-head-field, equilateral, that with a 10th and a 30th torn out, stroke steps of ditto and steps of 26 go. What this means is that for an equilateral triangle with the side s the area A is A = 5/2 - (sqs. 3)/2 - s = (appr.) s/2 - (1 - 1/10 1/30) - s metro-mathematical “field-side-and-area texts” from the Old Akkadian period (c. 2340-2200 BC) contain relatively complicated computations of side lengths, but no illus- CS areas of squares with given trating diagrams (see Fri- / berg, MSCT 1, Sec. A6.2; : id., CDLJ 2005: 2, 88 4.3- ; 4.7). TSS 77 is a fragment of Y \ | \ dla A j i a round tablet with a dialn o” gram of a square with four Fig. 3.1. TSS 77. A dia- (Thus, in terms of common fractions, the Old and inscribed circles (see Frigram on a fragment of a Late-Babylonian approximations to the square side of berg, Amazing Traces, 3.1 round tablet from Old 3 are, respectively 6.2.1; Fig. 3.1 below). The Babylonian Kisurra. = ;30 - (1 - ,08) - sq. s = ;26 : sq. s.

information that this fragment is from Old Babylonian drawing of a square on Y BC 7289 has been published Kisurra, and not from Early dynastic Suruppak, as was before by the present author, for instance in Amazing commonly believed earlier, is due to Krebernik, NABU Traces, Fig. 16.7.2. See now also Robson, MAI, 111, 2006: no. 15. Fig. 4.8. A square with its diagonals is depicted on the Old A geometrical doodle on the back of an administra- Babylonian tablet YBC 7289 (Friberg, MSCT 1, Fig. tive list from Old Babylonian Mari has the form of a 16.7.2; Fig. 3.2 below). An accurate approximation is square divided into 16 smaller squares, all with diagoused for the computation of the length of the diagonal nals. See Fig. 1.1, bottom. (Ziegler 1999, no. 37.) when the side of the square has the length 30. The way MS 3050 (Friberg, MSCT 1, Fig. 8.2.2; Fig. 3.3 in which this accurate approximation can have been below, left) is a round tablet featuring a square with obtained is discussed in Friberg, Amazing Traces, 397. diagonals, inscribed in a circle. It is hard to make The copy of YBC 7289 first published by Neugesense of the scattered numbers recorded inside and bauer and Sachs in MCT, 42, shows the square standoutside the diagram. It is likely, however, that the text ing on one of its corners, with the diagonals horizontal is the result of a school boy’s attempt to compute the and vertical. Subsequently, the same copy has been areas of the various parts into which the circle is republished on numerous occasions in various books divided by the square when, as usual, the length of the and papers, with the square oriented in this way. This circumference of the circle is given. is unfortunate, since orienting a square like this is in problem #37 in the demotic mathematical papyrus Interestingly, violation of an easily observed convention in Old P. Cairo (Friberg, Unexpected Links, Sec. 3.1 k) is of Babylonian mathematical texts, according to which precisely this kind, except that the length of the direpresentations of triangles, squares, rectangles, trapeameter, rather than the circumference, is given. zoids, etc., always are oriented with one side, the Two approximations to sqs. 2 (the square-side, ‘front’ or ‘upper front,’ facing left, as for instance, the alternatively square root of 2), 1;25 and 1;24 51 10, triangle in Fig. 2.1 above, as well as the hexagon and are mentioned in two Old Babylonian tables of conthe heptagon in Fig. 5.1 below. (Note, however, that stants (TMS 3 and NSe = YBC 7243), in the following this Old Babylonian convention does not apply in the way: case of the hexagon in the Neo-Sumerian text MS 1 25 constant of the diagonal of a 1 24 SI 10 the diagonal of an equalside square 1983/2 (Fig. 5.2 below). The correct orientation of the TMS 3 31 NSe 10 The accurate approximation sqs. 2 = 1;24 51 10 is obv. explicitly mentioned also in YBC 7289 (Fig. 3.2 above), where it 1s inscribed along the diagonal of the square in the diagram, and where it is used to compute the diagonal d of a square with a side of length 30: d = 1;24 51 10 - 30 = 42;25 35. The value 42 25 35 is recorded just below the diagonal in the diagram. Fig. 3.2. YBC 7289. An Old Babylonian tablet showing a square and its diagonals. obv. # 40 # 36 I us 1b.s1g Sa.ba 4 sag.du 1°6 gan gis.mà.gurg Sgán a.$a.bı gestà za. / rg mi en. pai Fig. 3.3. MS 3050. A square with diagonals inscribed in a Fig. 3.4. BM 15285 ## 36 and 40. A square divided circle. into various pieces. What is the area of each piece?

It is interesting that 1;24 51 10 is, essentially, the ing the well known Old Babylonian rule for the com- Same accurate approximation to sqs. 2 as the one used putation of the area of a circle, he could compute the in Ptolemy’s Syntaxis area of the concave square as Traces, Sec. 16.4, 1.10! (See Friberg, Amazing and Heath, HGM II, 276-278.) Indeed, the preliminaries to the Table of Chords in Book 1.10 of Ptolemy’s Syntaxis or the Almagest (150 A(concave square) = sq. 30 - 4 : 1/4 - (05 - sq. (3 - 30) = sq. 30 — ;45 - sq. 30 = 315 - sq. 30 = 3345. The story does not end with the Old Babylonian AD) include the computation of the side of a regular text BM polygon inscribed in a circle, expressed as a multiple Slotsky has published the Neo-Babylonian tablet BM of the 120th part of the diameter of the circle, when 47431 (Fig. 3.5 below) with a diagram on the obverse 15285 #36. Indeed, Robson in Festschrift the regular polygon in question has 10, 5, 6, 4, or 3 showing four circles inscribed in a square, and with a sides. (This is equivalent to computing the chords of brief text on the reverse giving an explicit answer to 36°, 72°, 60°, 90°, and 120°.) One of the approximaan (unstated) problem of the same type as the explictions mentioned by Ptolemy is itly stated problem in BM 15285 # 36. sqs. 7200 = 84;51 10. In spite of the apparent similarity, there are pro- Since, 7200 = 2 : 3600 = 2 - sq. 60, the corresponding nounced differences between the Old Babylonian text accurate approximation to sqs. 2 1s BM sqs. 2 = 84;51 10 / 60 = 1;24 51 The diagram of a square with four inscribed circles in the Early Dynastic text TSS 77 (Fig. 3.1 above) reappears in the well known Old Babylonian geometric theme text BM 15285, in one of 41 exercises where, in each case, a square with the side 1 (: 60) is divided into several pieces by a number of straight or curved lines, and the goal of the exercise 1s to compute the areas of all the pieces (see Friberg, Amazing Traces, Figs. 6.2.2-6.2.3; Robson, MAI, Fig. 2.10). Although the statement of the problem 15285 #36, and the Neo-Babylonian text BM 47431. Thus, while the side of the square in the former 10. in BM 15285 # 36 is lost, it is clear that what is asked for is the area of each small piece of the divided square: four circles, one ‘ear-of-sammii’ (a “concave square’), four half concave squares and four quarter concave squares, text is 1 - 60 ninda (1 ninda or ‘rod’ = c. 6 m), the side of the square in the latter text is 1 - 60 cubits (1 cubit = c. 1/2 m). Moreover, while the sizes of the pieces in the former case were supposed to be expressed in terms of area measure, the sizes of the pieces in the latter case are given in terms of “common seed measure’ (see Friberg, BaM 28, Sec. 1 and Sec. 6 b). The arbitrarily fixed relation between area measure and common seed measure (csm), expressed either as the amount of seed nominally needed to seed a certain unit of area or, conversely, as the area seeded by a certain capacity unit of seed, can be expressed in various ways, for instance as follows: 1 panu (pi) of seed (csm) corresponds to 3 : sq. (1 : 60 cubits), or in the case when the side of the whole square is given sq. (1 as 1: 60 (ninda). (csm). Fortunately, the corresponding statement in # 40 of a more complicated variant of the same problem is fairly well preserved. No solutions are offered in the text to the stated : 60 cubits) corresponds to 2 sutu (ban) of seed The following factor diagram for the Neo-Babylonian system of capacity measure shows how various units of that system were related to each other: problems in BM 15285. In the case of problem # 36 it is impossible to know if the school boy who was obv. asked to find the answer to the problem was supposed to use entries from a geometric table of constants, or if he was supposed scratch. to In $e.n. or Se.numun 'seed' n ZanteU6. sa. du us.sa.du ‘surrounding’ 47 sila ni \4- ta kip- pat kippatu 'circle' 1 2' sila Seln. 4 sag.dù pa-tay patru 'dagger start from 7? n. Sen. h sag Sal- hu Salhu ‘outer wall' the latter | fiat 72 n. Se.n. un zà. 2' = 1/2, n. = ninda 'rod' case, he could find, for mi instance, the area of the LI . = pab.pab 2,, x$e.n. mes-hat Èan.za.mi 'sammü-field' central x se i TR a-na a- ma- ri Sa lú Sa-tii pab.pab 'total, sum' concave square as the area of the central small square minus the combined area as_ mes-hat 'size' a.sa 'field' of four small quarter cir- Fig. 3.5. BM 47431. A square divided into various cles. In other words, uspieces. What are the seed measures of the pieces?

basket c. 1 liter piece of bread C(NB): panu I sam << qui — akalu ban pi ninda sila Therefore, the answer on the reverse of BM 47431 Fig. 4.1. VA 5953. An Old to the (unstated) question can have been computed in Babylonian mold showing a number of steps, as follows (cf. Robson, Festschrift five entangled bearded men Slotsky, 219-220): forming a pentagram. 1. The common seed measure of the square field surrounding the circles is 2 ban (csm) 2. The diameter of each one of the inscribed circles is 30 cubits The circumference of each one the 4 circles is (appr.) 3 - 30 cubits = 1 30 cubits The combined area of the 4 circles is (appr.) 4 - ;05 - sq. (1 30 cubits) = 45 00 sq. cubits The seed measure of the 4 circles is (appr.) ;45 - 2 ban = 1 1/2 ban = 1 ban 3 sila rev. line 2 3. The quarter-arc of each one of the circles is 1/4 - 1 30 cubits = 22;30 cubits obv., diagram The area of a concave square with this arc is (appr.) ;26 40 - sq. (22;30 cubits) = 3 45 sq. cubits The corresponding seed measure is (appr.) ;03 45 - 2 ban = 7 1/2 ninda 4. The seed measure of the 4 ‘dagger’-like concave triangles is 4 : 1/2 - 7 1/2 ninda = 1 1/2 sila rev. line 3 5. The seed measure of the 4 ‘outer-wall’ concave triangles is 4 * 1/4 : 7 1/2 ninda = 7 1/2 ninda rev. line 4 6. The seed measure of the central concave square is 7 1/2 ninda rev. line 5 7. The total seed measure is 1 ban 3 sila + 1 rev. line 6 1/2 sila + 7 1/2 ninda + 7 1/2 ninda = 2 ban Note the use in these computations of the following all of which have one side of length 60 and two sides well known ‘constants’ (igi.gub): of length 50. The area of one such triangle is 30 - 40 5 the ‘constant for a circle’ 26 40 the ‘constant for an (ear-of-)sammü-field (con- = 20 (- 60). Hence the area of the 5-front is 1 40 (- 60). cave square). There is no other known occurrence of a regular pentagon in a Mesopotamian text. However, VA 5953 The constant 5 for a circle appears in 7 Old Babylonian (Friberg, Amazing Traces, Fig. 7.9.7; see Fig. 4.1 tables of constants (see Robson, Mesopotamian Matheabove) is an Old Babylonian mold showing in relief matics, Sec. 3.1) and also in the Neo-Babylonian table five entangled bearded men forming a 5-sided star of constants CBS figure (a pentagram) enclosing a regular pentagon. 10996 (see Sec. 11 below). The constant 26 40 for a concave square appears in 4 Old A much older Mesopotamian example of a picture Babylonian tables of constants (see Robson, Mesopoof a pentagram is UE 3, 398 (Friberg, Amazing Traces, tamian Mathematics, Sec. 3.7). It is likely that it also Fig. 7.9.2; Fig 4.2 below), a copy of a seal imprint appeared in the now lost part of the Neo-Babylonian from a layer beneath the royal cemetery at Ur, dated to table of constants CBS 10996 (Sec. 11 below). the proto-Sumerian Jemdet Nasr period around the beginning of the 3rd millennium BC. Note the appearance in the lower left corner of a pentagram drawn in 4.n=5: Regular Pentagons and Pentagrams one uninterrupted line. An entry in TMS 3, an Old Babylonian table of constants from the ancient city Susa (in western Iran), mentions in the following way the ‘constant’ for a ‘5front’ (a regular pentagon), meaning the area of the 5front when the side length is 1 (- 60): 140 igi.gub 1 40 the constant of a S-front Sa sag. 5 TMS 3 26 The value 1 40 is easily explained: If the side length of the pentagon is 60, then the length of the circumscribed circle is (appr.) 5 ‘ 60. Therefore, the radius of the circumscribed circle is (appr.) 5 : 10 = 50. Consequently, the 5-front can be divided into five triangles, Fig. 4.2. UE 3, 398. A pentagram appearing in a seal imprint from the proto-Sumerian Jemdet Nasr period.

Actually, a pentagram appears in this seal imprint sooner for calligraphic than for artistic reasons. Indeed, in the proto-cuneiform script used in Mesopotamia in the Jemdet Nasr period, the sign UB had the form of a pentagram. (Several of the other images in the seal imprint in Fig. 4.2 are also proto-cuneiform signs.) Just as in this seal imprint, UB appears frequently in proto-cuneiform texts from Jemdet Nasr together with the sign AB. An example, borrowed from Englund and Grégoire, MSVO 1, is shown in Fig. 4.3. Fig. 4.3. Englund and Gregoire, MSVO |, 220 = IM 55587. A proto-cuneiform text from the proto-Sumerian Jemdet Nasr period. 5. n = 6: Regular Hexagons Another entry in the Old Babylonian table of constants TMS 3 mentions the ‘constant’ for a ‘6-front’ (a the image of a regular hexagon with a circle in the middle, probably some kind of geometric assignment. regular hexagon), meaning the area A of the 6-front when the side length is 1 (- 60): 2 37 30 igi.gub 2 37 30 the constant of a 6-front sa sag.6 TMS 3 27 In view of the discussion above of the case n = 3, this value can be explained as A=6-1/2-(1 - 1/8) ‘ sq. 60 = 6 - 26 15= 2 37 30. A regular hexagon with sides of length 30 is depicted on the obverse of TMS 2 (Friberg, Fig. MSCT 1, 8.2.15; Fig. 5.1 below), an Old Babylonian tablet from Susa. As indicated by the number 6 Fig. 5.1. TMS 2. An Old Babylonian tablet from 33 45 recorded in the Susa with images of a hexagon and a heptagon. left-most equilateral sub-triangle, the area of a regular hexaobv. gram with this side length could be computed as follows: A=6- 1/2 + (1 - 1/8) - sq. 30 = 6 : 26 15 - 1/4 = 6 - 6 33345 (= 39 22;30). MS 1983/2 (Friberg, MSCT 1, Figs. 8.1.12, 8.2.14; Fig. 5.2 below) is a large fragment of a mathematical tablet, probably from the Neo-Sumerian Ur III period. The tablet is inscribed on the obverse with a diagram showing a trapezoidal field divided into five parallel stripes with areas forming an arithmetic progression, and on the obverse (ac- Fig. 5.2. MS 1983/2. A mathematical tablet, probably cording to a likely reconstruction) with from the Neo-Sumerian Ur III period.

6. n = 7: Regular Heptagons and Two Kinds of eral crucial words in the text and an interesting new interpretation of the whole text. The discussion below 7-Sided Star Figures of the text is largely based on their new interpretation. A third entry in the Old Babylonian table of con- The reverse of the tablet is completely destroyed. stants TMS 3 mentions the ‘constant’ for a ‘7-front’ (a On the obverse, in a box in the upper left corner, regular heptagon), meaning the area A of the heptagon which is fairly well preserved, there is a diagram of a when the side length is 1 (+ 60): 7-sided star figure (a heptagram), drawn in one unin- 341 igi.gub sa 3 41 the constant of a 7-front terrupted line consisting of a chain of 7 straight lines sag.7 TMS 3 28 There is also an image of a regular heptagon on the reverse of TMS 2, the tablet from Susa shown above in Fig. 5.1. The heptagon has the given side length 30, and therefore the radius r of the circumscribed circle can be computed as of equal length), and in the lower half of the obverse there is a numerical table. The rest is empty. By luck, although there are some missing parts of the clay tablet, nothing seems to have been lost of the inscription on the lower half of obverse. See Fig. 6.1 below. The star figure on the obverse of CBS 1766 is inscribed in a double circle. Both the star figure and r = (appr.) 1/6 : 7-30 = 35. That is, of course, why the value 35 u$ ‘35, the length’ is recorded above a radius of the heptagon on the the circles are drawn without much care, and without the use of compass and ruler. The seven points of the star figure are numbered, from 1 to 7, and briefly reverse of TMS 2. One would now expect to find the total area of the labelled in the following way: 1 qu-ud-mu the foremost 2 Tsal-mu-Sum the next 3 [Sal-Su qat-nu] [the third, thin] 4 e-ba-nu the one constructed by (the god) [nigin sag sa] sag.7 5 ha-an-su the fifth a.na 4 te-si-ip-ma 6 re-'bi uh-ri the fourth behind upper triangle or of the whole heptagon recorded in the diagram. That is not the case. Instead one finds a somewhat cryptic inscription, interpreted as follows by Robson in Mesopotamian Mathematics, 49: Ea Si-in-Sé-ra-ti ta-na-as-sa-ah-ma a.da the third behind. [The square of the front] (the side) of the 7-front by 4 you repeat, then These are known names for seven strings of the Mesothe twelfth you tear out, then the field (the area). potamian harp or lyre. See the further discussion of What this means is that the area of a heptagon with the side s can be computed as A = (appr.) 4 : sq. s — 1/12 of 4 - sq. s = 4° sq. s — 320 - sq. s = 3540 : sq. s. this topic in the 10th section of the present paper. Note: In Babylonian mathematical texts, the essential components of geometric figures are their straight or curved segments, never their vertices. Cf. the discussion of Babylonian “metric algebra diagrams” vs. Greek Thus, you get the area of the heptagon if you multiply “lettered diagrams’ in Friberg, Amazing Traces, Sec. the square of the front by 4, and reduce the result by 1.1. Therefore, it is likely that the numbers inscribed a twelfth of its value. The computation rule is a handy around the star figure in CBS 1766 should be undervariant of the more formal rule A = sq. s © 3;40. stood as components of number pairs defining the Compare the entry ‘3 41 the constant of a 7-front’ in seven sides of the star figure, not as single numbers TMS 3 28, where the value 3 41 probably had been defining the seven points of the star figure! computed as follows. When s = 1 (: 60), then A = (appr.) 7 : 30 - sqs. (sq. 1 10 - sq. 30) = 7 : 30 - 20 - sqs. 10 = (appr.) 7 : 10 : 3 10 = (appr.) 3 41 ( 60). CBS 1766. Description of the Diagram and the Table A photo of CBS 1766 was first published by The table on the obverse of CBS 1766 contains 1] columns, organized as follows: Column 1 is empty, columns 2-3 are both inscribed with 7 lines of number pairs. Column 4 is again empty, while columns 5-6 are inscribed with only 1 line each of number pairs. Apparently, the writing of numbers in the table was Hilprecht in his Explorations on p. 530, where the text interrupted here and never finished. Indeed, the rewas loosely characterized as an “astronomical tablet maining columns are empty, except for the last colfrom the Temple Library.” Subsequently, CBS 1766 was largely ignored for umn, which contains traces of a few words (not nummore than a century, until Horowitz saved it from but another possibility is that the text in its present oblivion by republishing a photo of it in JANES 30 pp. 37-53, together with a transliteration of the text and an been asked to fill in the remaining numerical parts of attempted interpretation of it. the table, which he never did. bers). The interruption may have been unintentional, state was an assignment, and that a school boy had A year later, in NABU 2007/2, C. Waerzeggers and There is a (somewhat) readable line of text as a R. Siebes suggested alternative transliterations of sevheading over columns 1-4 in the table. The headings

CBS 1766. A Geometric Explanation of the Number Pairs in the Table In column 2, the first inscribed column on CBS 1766, the second number in each pair is equal to the first number in the next pair. Thus, the first pair 2, 6 is followed by . the second pair 6, 3, the + , - nie mur. Tie > E5 SE DR er £eo $- be + pro: ~~, ant È peri .è =à , 4 RT 4 cu a AVx«a]na:“y.et 3=- x2,=n Prin ~way eres gie CIO third pair 3, 7, and so on. uLE To make sense of this observation, assume that In column 2, the pair 2, stands for the side 6 in the 7-sided star figure which 1 goes from the point rote labelled 2 to the point labelled 6, the next pair 6, 3 stands for mm I mm mm x the side of the star È belled 6 to the point figure from which the goes point lalabelled 3, and so on. See Fig. 6.2, top. Thus, o the seven number pairs in column 2 can be inter- | IM iden etm) ir | x preted as A description of how >| +. | | ls al2í kaka] ! 7 4 le 5 | I 4 Ta 1 XXX to draw the whole 7- | sided star figure by Sar-tum i- | kiAt-mu-um | | i E XXX mulbi.im | eae \LI 1 B 2 5176 eat EEDS | | use of an uninterof chain rupted straight lines running through the points labelled 2, 6, 3, 7, 4, 1, 5, 2, in this order. Now, correct if this is the interpretation of the seven number pairs in the first inscribed col- Fig. 6.1. CBS 1766. A 7-sided star figure and a numerical table. Photo and conform umn on CBS 1766, what transliteration. Published here with the kind permission of G. Frame. is then the corresponding interpretation of the seven over the other columns of the table, if any, are unreadnumber pairs in the second inscribed column, column able, except for the heading over the non-numerical 3? It ought to be entries ın the last column, but that too is badly readThe line of text above columns A description of how to draw a certain diagrammatic figure by use of a set of (non-connected) straight lines able. 1-4 is read as follows by Horowitz, JANES 30, pp. 37-53: IM si-im-da-tum zi-gi-pu ig-r[i-bu? ...] or ik-ta[l-du? ...] pairs of altitudes which appro[ached ...] or rea[ched ...]. running through the pairs of points labelled (1, 7), (5, 4), (2, 1), (6, 5), (3, 2), (7, 6), and (4, 3). See again Fig. 6.2, top. This diagrammatic figure is clearly a regular heptagon, that is a polygon with 7 equal sides, which can be inscribed in a No serious attempt was made by Horowitz to explain the meaning of this line of text. circle. Assuming that this interpretation is correct, it remains to explain what the precise relation is between

consecutive parallel, nearby sides of the sides of the 7/3 star figure heptagon 2, 6 1,7 sided star figure running through the points 2, 6, 3, 7, 4, 1, 5, 2. In a similar sense, the heptagon itself can be understood as a “7/1 star figure.” 6,3 5,4 This observation immediately leads to the fol- 3,7 2.1 lowing question: What is then a “7/2 star figure?” 7,4 6,5 4,1 3,2 LS 7,6 3,2 4,3 The answer is demonstrated by the diagram in Fig. 6.2, bottom. Start, say, at the point 7 and proceed from there counter-clockwise to the 2nd point along the circle, which is then the point 2, and draw the straight line from 7 to 2. Repeat the parallel, distant consecutive process until the diagram returns to the starting sides of the sides of the heptagon 7/2 star figure 5,4 7,2 point. The result will be a new kind of 7-sided star figure running through the points 7, 2, 4, 6, 1, 3, 5, 7, in this order. Now observe that the side [7, 6] [2, 1] [2, 4] [4, 6] [4, 3] [6, 1] [6, 5] [1, 3] this observation explains the two pairs 5, 4 and 7, [1, 7] [3, 5] 2 in the first line of columns 5-6 on CBS 1766. [3, 2] [S, 7] Fig. 6.2. Suggested geometric explanation of the four preserved tables of number pairs on CBS 1766. from 5 to 4 in the heptagon is parallel to the side from 7 to 2 in the 7/2 star figure. It is likely that Consequently, it is likely that the unfinished second pair of inscribed columns was intended to show, quite explicitly, that For each side in the 7/2 star figure there is a parallel the two number pairs in each line of columns 2-3. In side in the heptagon going in the same direction. particular, what is in the case of the first line of the See again Fig. 6.2, bottom. (It is not clear why the pair two columns the relation between the side from 2 to 6 5, 4 precedes the pair 7, 2, so that the order of the two in the star figure and the side from columns is not the same in the case of the 7/2 star 1 to 7 in the heptagon? The answer is obvious, since the two sides figure as in the case of the 7/3 star figure.) are parallel to each other and go in the same direction. Since the numerical table on CBS 1766 was left Similarly, in the case of the second line, the side from unfinished, it is impossible to know what a third pair 6 to 3 in the star figure is parallel to the side from 5 of inscribed columns (columns 8-9) could have conto 4 in the heptagon and goes in the same direction. tained. Maybe they would have been concerned with a And so on. (It is assumed here that, like the star figure, “7/4 star figure.” Now, it is easy to see that a 7/4 star the heptagon is drawn in one uninterrupted chain of figure is identical with a 7/3 star figure running in the straight lines, counter-clockwise, so that each side of opposite direction, through the points 2, 5, 1, 4, 7, 3, the heptagon has a given direction.) Therefore, the 6, 2. Similarly, a 7/5 star figure is just a 7/2 star figure number pairs in the first two inscribed columns on running in the opposite direction, through the points 7, 5, 3, 1, 6, 4, 2, 7, and a 7/6 star figure is the same as CBS 1766 quite explicitly demonstrate that For each side in the 7-sided star figure there is a a heptagon running in the opposite direction, through parallel side in the heptagon going in the same directhe points 1, 2, 3, 4, 5, 6, 7, 1. tion. It is now time to return to the meaning of the line Note that because of its manner of construction, the 7-sided star figure, like the regular heptagon, can be inscribed in a circle, that is, both figures are “cyclic.” The diagram on the obverse of CBS 1766 shows the 7- sided star figure being inscribed in a double circle, but that 1s probably just an embellishment (if not to accommodate further text). of text over columns 1-4. It is suggested here, very tentatively, but in essential agreement with one of the possibilities suggested by “7/3 star figure.” What this means is that in order to that there are over into column 4), and that these headings should be read as, respectively, col. 1: "IM! In Fig. 6.2, top, the 7-sided star figure is called a Horowitz, three distinct headings in columns 1, 2, and 3 (spilling directions’ col. 2: si-im-da-tum pairs col. 3: zi-gi-pu iq-r[i-hu] the stakes are close together. of the The “pairs” mentioned in the heading over col. 2 numbered points, say the point 2, proceed from there can, of course, be understood as the number pairs from counter-clockwise to the 3rd point along the circle, 2, 6 to 4, 2 specifying the 7 sides of the 7/3 star figure. draw the star figure, you can start at one which is then the point 6, draw a straight line from 2 Finally, since stakes are usually straight (and upto 6, and then repeat the procedure until the diagram right), it is possible that in this text the term ‘stakes’ returns to the starting point. The result will be the 7- stands for ‘straight lines,’ and that the meaning of the

drawn in one uninterrupted line, and its 4 diagonals. A previously unpublished Old Babylonian tablet from Haddad (Fig. 7.2) is inscribed with an 8/2 star figure and its 4 diameters. The 8/2 star figure cannot be drawn with one uninterrupted line. Instead it 1s composed of 2 squares. The meaning of the many scrib- Photo: F Al-Rawi bled cuneiform signs inside the figure on the obverse of this tablet, Fig. 7.1. IM 51979. An Old Babylonian(?) tablet and similarly scribbled signs on the showing an 8/3 star figure with its diagonals. reverse, is not at all clear. obv. 8. n = 12: A 12-sided Star Figure in a Seleucid Astrological Text The Seleucid astrological text O 176, in which a 12-sided (and 12- pointed) star figure appears in an isolated position, was first published by Thureau-Dangin in TCL 6, text 13. A commentary appeared much later, in Rochberg-Halton, ZA 77. Names of months and planets are Copy and photo: F. Al-Rawi inscribed in the 12 points of the star figure (see Fig. 8.1 Fig. 7.2. An Old Babylonian tablet from Haddad with an 8/2 star figure and scribbled cuneiform signs. line of text over column 3 below). However, there is no obvious connection is that the straight lines making up the sides of the 7/3 star figure are nearby the straight lines parallel to them, which make up the sides of the heptagon, and which are specified by the number pairs in column 3. See Fig. 6.2, top. In contrast to the situation in Fig. 6.2, top, the situation in Fig. 6.2, bottom, is that the straight lines making up the sides of the heptagon are distant from the straight lines parallel to them, which make up the sides of the 7/2 star figure, and which are specified by the number pairs in column 6. 7. n = 8: Two Old Babylonian? Tablets (Abbreviated) month names: BAR (1) SU (IV) DU, (VII) AB (X) GU, SIG, (II) (ID NE KIN (V) (VI) APIN (VIII) GAN (IX) ZIZ SE (XD (XII) with Two Kinds of 8-Sided Star Figures IM 51979 (Friberg, Amazing Traces, Fig. (Abbreviated) planet/god names: DIL.BAT (Venus/Ishtar) US (Saturn/Ninurta) GU, SAL (Mercury/Nabü) GENNA(?) (also Saturn?) (Mars/Nergal) 7.8.2; Fig. 7.1 above) is a roughly made tablet, Fig. 8.1. possibly Old Babylonian, inscribed exclusively circumscribed circles, month names and planet (or god) names. with a diagram showing an 8/3 star figure, The astrological meaning of this diagram is unknown. O 176. A twelve-sided star figure with inscribed and

between these names of months and planets on one on the reverse of the lexical text VAT 9128 from Early hand and the astrological text on the tablet on the Dynastic Illa Shuruppak, 2600-2500 BC. The photo of other hand. the reverse of the clay tablet in Fig. 9.1 below (avail- The 12-sided star figure can be more distinctly able online at cdli.ucla.edu, P010673) shows a last characterized as a /2/4 star figure, since each side in half-line of the lexical text, and as space fillers on the the star figure goes from one of the 12 points of the otherwise empty reverse a drawing of a grazing antestar to a point 4 steps removed from it along the circle. lope and a doodle in the form of a kind of non-regular The star figure cannot be drawn with one uninter- 6-sided star figure with embellished “diagonals.” The rupted line. Instead, it is composed of 4 equilateral way ın which the doodle was drawn is illustrated in triangles. Note the presence of both a circumscribed Fig. 9.2 below. and an inscribed circle. Two other examples of quasi-mathematical spacefilling doodles on the reverses of similar texts from Early Dynastic Illa Shuruppak are shown in Friberg, 9. A Starlike Doodle on the Reverse of an MSCT 1, Figs. A6.21-22. Early Dynastic IIa Lexical Text Loosely associated with the theme of polygons and Star figures on Mesopotamian clay tablets is a doodle 10. Names of Strings, Intervals, and Modes on an Instrument with 9 Strings For the readers’ convenience, in this section of the paper are brought together some well known facts about cuneiform texts mentioning names of strings, intervals, and scales on a harp or lyre with 9 strings. of such items is necessary A knowledge for the proper understanding of the meaning of the 7-sided star figure and the numerical table on CBS 1766. The names Sumerian and of the 9 Akkadian, strings, are in both mentioned in columns 1-11 of UET VI 126, a Neo-Babylonian fragment of a copy of the 32nd tablet of the lexical series Nabnitu ‘creation’ (Fig. 10.1). See Kilmer, Festschrift Landsberger. According strings are to this text, counted five of the nine from the front (of the string instrument), while the remaining strings are counted from the rear. Fig. 9.1. VAT 9128. The reverse of a lexical text from ED Illa Suruppak with a drawing and a doodle. The photo is reproduced here with the kind permission of Bildagentur für Kunst, Kultur und Geschichte. CRS TK Fig. 9.2. VAT 9128. The construction in three steps of the doodle.

$ 1 sa. di qud- mu- Fr sa. us Sa-mu-Su- Fr Sa-al-Su ga-attnu wg sa. 3. sa. sig sa. 4. a5! hatur ba-nu-hi, am| sa. Ki. sa.4.a.ga.gul! re-bi uh- ri- im sa.3.a.ga.gul' Sal-Si úh- ri- i FH PR Si-ni üh- ri- im da uhruum 9 pi- it! .a| . nu . piismu i- Sarti 10.1. VET VII 126, a fragment from Ur of a copy of the 32nd tablet of the lexical series Nabnitu. The copy is published here with the kind permission of the Trustees of the British Museum. UET VII 126, cols. i-ii § 1 sa.di qud-mu-u fore string [x x x] [si-h]i-ip i-Sar-tum sa.us sá-mu-Su-um next string [x x x] [ki-i]t-mu kitmu sa.3.sa.sig sa-al-Su ga-alt-nu] third, thin string [x x x] [si-hi-ip kli-it-mu sihip kitmi sa.4.tur a-ba-nu-[u] fourth, small string [x x x] [em-bu-bu]-um embubu sihip isarti / Ea-created sa.k1.5 ha-am-[Su] fifth string sa.4.a.ga.gul re-bi uh-ri-[im] fourth rear string sa.3.a.ga.gul Sal-si uh-ri-im third rear string sa.2.a.ga.gul Si-ni uh-ri-im second rear string sa.l.a.ga.gul uh-ru-um rear string [9] sa.a 9 pi-it-nu nine strings § 2 [sa.d]u.a [sa.si.s]a pi-is-mu 22? i-Sar-ti isartu The names of 7 of the 9 strings reappear in the Neo-Babylonian table of constants CBS 10996, col. vi (Kilmer, Or 29; see section 11 below), together with names for two alternating (dichords) each. CBS 10996, obv., col. vi [1 5 sa nis tuh-ri] ‘rise of heel(?)’ [7 5 sa se-e-ru] ‘song’ [2 6 sa i-Sar-tu,] ‘normal’ [1 6 sa Sal-sa-tu,] ‘third’ [3 7 sa em-bu-bu] ‘reed-pipe’ [2 7 sa 4-tu] ‘4th’ [4 1 sa Sub.murub,] ‘fall of middle’ [1 3 sa gis.Sub.ba] ‘lot, share’ [5 2 sa murub,-tu,] ‘middle’ 2 4 [sa ti-tur] murub,-tu, ‘bridge, middle’ 6 3 sa kit-mu ‘cover’ 3 5 sa ti-tur i-Sar-tu, ‘bridge, normal’ 74 sa pi-tu, ‘opening’ 46 sa ser-du sa qud-mu-u u sa 5-Su sa 3° uh-ri u sa 5-Su sa Sa-ge, u sa 4 uh-ri sa qud-mu-u u sa 4 uh-ri sa 3-$u sig u sa 3-Su uh-ri sa Sa-ge, u sa 3-su uh-ri sa de-a.dü u sa qud-mu-u sa qud-mu-u u sa 3-$u sig sa x 5-SU u sa Sa-ge, sa Sa-ge, u sa “é-a.dú ‘lament’ BNUYWES PLtD©JAÀ=BUi sa nis tuh-ri u means ‘and’ sa Se-e-ru sa i-Sar-tu, sa-ge, = Sa-musu sa Sal-Sa-tu, sa em-bu-bu sig = gatnu ‘thin’ sa 4-tu sa Sub.murub, du = banü ‘created’ sa giS.Sub.ba sa murub,-tu sa ti-tur murub,-tu murub,-tu = gablitu ‘middle’ sets string pairs

sa 4 uh-ri u sa 3-su sig 6 3 sa 3-su sig u sa 5-Su [3 5 sa 3-su uh-ri u sa [‘é-a.dù 74 sa pi-tu,] sa “é-a.du u [sa 4 uh-ri 4 sa ser-du] The column 6 sa [kit-mu] sa ti-tur i-sar-tu,] (partly reconstructed and corrected Note, finally, that in col. vi of CBS 10996, the first above) begins with a brief version of the list of 2 times set of dichords is ordered /exicographically rather than 7 dichords in terms of only the numbers of the strings, in the order corresponding to successive sides of the but then, as an afterthought, repeats the list with the 7/3 star figure. More precisely, the 7 dichords in the dichords expressed also in terms of the full names of first set succeed each other in the following way: the strings. 1, The names given for the dichords remain to thıs day largely unexplained. (The reading tuh-ri ‘heel, Achilles tendon?’ is due to Mirelman and Krispijn, Iraq 71.) In Fig. 10.2 below, top and bottom, one “primary” set of 7 dichords is identified with the seven sides of 5 2,6 = 1,5 + 1, 3, 7 = 2, 6 + 1,1 1 4,1 = 3, 7 + 1, 5, 2 = 4,1 + 1,1 6, 3 = 5,2 + 1, 1 7,4 = 6,3 + 1, 1 1 (8 = 1 mod 7) Therefore the 7 sides of the 7/3 star figure corresponda 7/3 star figure, while a “secondary” set of 7 dichords ing to successive primary dichords are obtained from is identified with the seven sides of a 7/2 star figure. each other through repeated rotation by 1/7 of a full Note that while this way of visualizing the two sets of revolution. 7 dichords mentioned in col. i of CBS 10996 is not known from any Babylonian text, it still is, of course, Similarly, the 7 dichords in the second set succeed each other in the following way: inspired by the diagram and the table on CBS 1766. 5, Note also that in Fig. 10.2, below, the sides of the 6, 1* = 5, 7 + 1, 1 7/2 star figures are oriented in the same way as in the 7, 2* = 6, 1 + 1, 1 7* In three 1, 3 = 7, 2 + 1,1 cases, marked by asterisks, the two numbers defining 2, 4 = 1,3 + 1, 1 3, 5 = 2,4 + 1, 1 4, 6 = 35 + 1,1 corresponding diagram in Fig. 6.2 above. a dichord correspond to a side in the 7/2 star figure with an opposite direction. This appears to be a mistake made by the author of the text. (8 = 1 mod 7) (8 = 1 mod 7) Therefore also the 7 sides of the 7/2 star figure corresponding to successive secondary dichords are obtained through repeated dmü queme po 1,5 rotation of the first side by 1/7 of a full nis tuhri revolution. _ Note: In Vitale, UF 14, 254, another way 230 alah of visualizing the two sets of 7 dichords 3,7 embiibu mentioned in col. 1 of CBS 10996 by use of 41 nid qabli (Sub.murub) 7/2 and 7/3 star figures is only superficially related to the method of presentation in Fig. 5; 2 qablitu (murub-tu ,) 10.2 above, which is based on the testi- 6,3 Si mony of the text CBS 1766. Vitale was, of 7,4 pitu course, unaware of the existence of CBS 1766. A secondary an eS ON > SP 5, 7 ” 6, 1* A, 2 5 n as 8 7 x N & > N e, ? MEL * key text for the understanding of Babylonian music theory is UET VII 74+ —_— f?r (Gurney, /rag 56; Dumbrill, Archaeomusicology, 48), a small fragment of an Old Yalÿatu Babylonian text with two explicit modal _ retuning algorithms for a string instrument 7, 2° rebütu (44u) L3 isqu (gi3.Sub.ba) 2 4 titur gablitu 3, 5 titur iSartu 4,6 serdu with nine strings (see Fig. 10.3). A second, newly identified fragment of the same text I);is UET ° (although not the same clay tablet!) VI/3 899, Mirelman and Krispijn, /rag 71. re The upper half of col i of UET VII 74+ contains what may be $ 1 of the text, apparently devoted to some kind of enumeration Fig. 10.2. A visualization of the two sets of 7 string of string pairs. It is difficult to say precisely pairs (dichords) mentioned in CBS 10996. how that first paragraph was organized.

In the transliteration below of the text of the fragment, missing parts of the text have been tentatively reconstructed, within straight brackets. UET VII 74+, col. ii 82,5 § 2.6 $ 2.7 end $ 3.1 $ 3.2 [Sum-ma **zà.mi pi-tum] / [If the sammü (instrument) is pitu,] [e-e]m-b[u-bu-um la za-ku] / the embubu is unclear, sa-al-S[a-am qa-at-na-am ta-na-sa-ah-ma] / the third, thin, you shall tighten, then e-em-bu-bu-[um iz-za-ku] / the embubu will be clear. sum-ma *z[a.mí e-em-bu-bu-um] / If the sammü is embubu, ki-it-mu-um [la za-ku] / the kitmu is unclear, re-bé uh-ri-im [ta-na-sa-ah-ma] / the fourth rear you shall tighten, then ki-it-mu-um iz-[za-ku] / the kitmu will be clear. Sum-ma **zà.mi k[i-it-mu-um] / If the sammü is kitmu, i-Sar-tum la za-[ka-at] / the isartu is unclear, Sa-mu-Sa-am u uh-ri-a-a[m ta-na-sà-ah-ma] / the samussu and the rear you shall tighten, i-Sar-tum iz-za-[ku] / then the isartu will be clear. nu-su-[hu-um] / Tightening. sum-ma ®za.mi i-Sar-t[um] / If the sammü is isartu, qa-ab-li-ta-am <la za-ku-ta-am> ta-al-pu-[ut] / the gablitu <unclear> you played, [S]a-mu-Sa-am u uh-ri-a-am te-[né-e-am-ma] / the samussu and the rear you shall loosen, [e°z]a.mi ki-it-mu-[um] / then the sammü will be kitmu. [Sum]-ma ®za.mi ki-it-m[u-um] / If the sammü is kitmu, [i-Sa]r-ta-am la za-ku-ta-am t[a-al-pu-ut] / the isartu unclear you will play, then [re-bi] uh-ri-im te-né-e-[am-ma] / the fourth rear you shall loosen, then [*8za.mi e-em-bu-bu-um] / the sammü will be embubu. By luck, the fragment UET VII 74+ contains both plained as follows: the end of one modal retuning algorithm and the § 3.1. If the string instrument is tuned to the isartu mode, and beginning of another. (A (recursive) algorithm is a the gablitu dichord is dissonant (‘unclear’), loosen the procedure in several steps, where each step is structur- Samussu (= second) string and the rear (= ninth) string. The string instrument becomes tuned to the kitmu mode. ally similar to the previous step.) § 3.2. If the string instrument is tuned to the kitmu mode, and The meaning of the preserved beginning of the the isartu dichord is dissonant, loosen the fourth rear second modal retuning algorithm can be vaguely ex- (= the sixth) string. The string instrument becomes tuned to the embubu mode. Evidently, as shown in Fig. $ 1 Lee ee" pr | ae [e-em-bu-bisuim iz-za- : sa re- UA |Sum-ma gi3.zàmi em-bu-bu- 4 $ 2.5 M fa al Àj-am qe at na am tu- na-sè- ah-ma > sa ga-ab-li tum 9 2 I tum _ Sa x /ga-ab- li- tim sa x/i-$artim ku um 10.4 below, the modes in this retunin g algorithm (after the obvious reconstruction) E we 82.6 the sides in the 7/3 star figure. | The application of the modal retuning algorithm in UET VII 74+, 83 sa Se ruum |Ki- it- mu-um_ id za- ___ Au: _ sa u _ ma gt, en | $2.7 sa se- er di- im |$a-mu-$a-am à úh- ri- alum tu-na-kà-ah-ma presupposes that the string instrument has been tuned already to some mode. me MS O A nal gener ae sa mode, Clearly ki- itmu-um ki- it- mu-um \la za-ku |re-bi üh-ri- im\ tu-na-sa-ah-ma | follow each other in the same order as iz- za- = si 5 | ANT gis.zà.mi i- Sarkum 7 a dae en |i-Sar-tum nu- su- \ga-ab-li-ta-am \ ku i ta- al- pu? ut ‚Sa-mu-Sa-am à uh-ri-a-am tené-e-ma Vz oF a = = — | ra NES la za-ku-ta-am ta-al-pu-ut nr, esi cas norma $31 cation of the following initial tuning ~ algorithm: After string 2 has been tuned in some arbitrary but appropriate way, = string 6 is tuned to make the isartu 93.2 Sn ascendina am. a sting 3 is o Pie e ısartu or mode, is easily obtained through appliifth). u xt, stri tuned to make the kitmu dichord 6, 3 ‘clear’ (a descending fourth). In the Fig. 10.3. UET VII 74, a fragment of an OB third step, string 7 is tuned to make the text from Ur with two retuning algorithms. embubu dichord 3, 7 ‘clear’ (an ascend-

quema . iSartu 6, 3 kitmu 3, 7 embübu . e RO, dv a = Tha a > = = No è MS > > $ A a, “Yop After six steps of this modal retuning algorithm in 2,6 an Y Ÿ und“ § 3 of UET VII 74+, all strings except string 5 have _ been ‘loosened.’ The initial interval of this mode is the gablitu 7,4 pitu 4, 1 nid gabli 1,5 nis tuhri I, 2 gablitu n . e dichord, and the nis tuhri dichord 1, 5 is ‘unclear.’ Therefore, this is called the gablitu mode. See the seventh diagram in Fig. 10.5. In the last step of the retuning algorithm, finally, string 5 is ‘loosened’ as well. Now the gablitu dichord is ‘unclear’ again, and the configuration is the same as in the Fig. 10.4. Dichords following each other as the sides of the 7/3 star figure. initial isartu mode, only with all strings ‘loosened.’ It is not only the retuning algorithm for seven modes in UET VII 74+, §3 that can be explained ing fifth). Then string 4 is tuned to make the pitu easily in terms of the 7/3 star figure in Fig. 10.4. Also dichord 7, 4 ‘clear’ (a descending fourth), string 1 is the varying distributions of tones and semitones in the tuned to make the nid gabli dichord 4, 1 Seven ‘clear’ (a modes can be explained without trouble by descending fourth), and string 5 is tuned to make the reference to the 7/3 star figure (although there is no nis tuhri dichord 1, 5 ‘clear’ (an ascending fifth). The known document indicating that the authors of the inevitable Babylonian texts discussed in the present paper were end result of this straightforward initial tuning algorithm is that the gablitu dichord 5, 2 becomes ‘unclear’ (in modern terms a disharmonic tritone, or more precisely, an augmented fourth, alternatively aware of this possibility). Consider, for instance, the 7/3 star figure for the isartu mode in Fig. 10.5. In that star figure, the a diminished fifth) and cannot be made ‘clear’ without dichord 2, 3 can be construed as a combination of the disturbing the given initial tuning of string 2. dichord 2, 6 (a descending fifth) and the dichord 6, 3 Both this assumed initial tuning algorithm and the (an ascending fourth). Therefore, the dichord 2, 3 can modal retuning algorithm in § 3 can be explained with be understood as a regular second (or rather a major reference to the 7/3 star figure in Fig. 10.4. As shown second, in modern notation a tone). In the isartu mode, in the first diagram in Fig. 10.5 below, in the generaother regular seconds (or tones) of the same kind are tive isartu ‘normal’ mode all the dichords correspond- 3, 4 and 4, 5, as well as 6, 7 and 7, 1. The situation ing to the sides of the 7/3 star figure are ‘clear’, except is different in the case of the dichord 5, 6, which can the gablitu dichord 5, 2. be construed as a combination of the dichord 5, 2 (an In the first step of the retuning algorithm in § 3, augmented ascending fourth) and the dichord 2, 6 (a string 2 is ‘loosened’ so that the gablitu dichord 5, 2 descending fifth). Therefore, the dichord 5, 6 can be becomes ‘clear.’ As a result, the isartu dichord 2, 6 called a minor second (a semitone). becomes ‘unclear,’ and the kitmu dichord 6, 3 becomes dichord 1, 2 can be understood as a combination of the Similarly, the the initial dichord of this new mode, therefore called dichord 1, 5 (a descending fifth) and the dichord 5, 2 the kitmu mode. See the second diagram in Fig. 10.6. (an augmented ascending fourth). Consequently, 5, 2 And so on. is another minor second (or semitone). In the same way, it can be isartu mode kitmu mode embubu mode 1 (8) pitu mode 1 (8) seen that in all the 7/3 star figures for the seven modes in Fig. 10.4, the ‘unclear’ dichord (dashed) is next to two minor seconds (semitones), while all the are major seconds (tones). In Fig. other seconds 10.5, the semitones are indicated by the letter s, while the tones are indicated by the letter f. In modern notations, the retuning algorithm in Fig. 10.5 can be expressed as follows: It is interesting that the result of applying the (partly hypothetical) Old Babylonian Fig. 10.5. The retuning algorithm in UET VII 74+, §3 retuning algorithms in terms of the 7/3 star diagram. (L = loosened.) sequences of descending fifths based on

B 3 7 4 1 5 2 1 2 3 4 5 6 7 A D G C FB C B A G F ED enge D) il v 7s Ss Ts Ss Ss 7s 68 os A D G C F Bb y a 7s Ss Ss A G + = + t Ss $— € F Bb F E D c bb ==“ to sot kitmu ing mode pitu ttt Ss Ts Ss The pitu instance, can A C Bb A G F Eb D sese ce AAA Ts 68 kitmu, augm. 4th t s t bb _ n embibu ge. s t string 4 so dichord that the becomes clear, then tuning string 1 ai Ts Ss embubu, 5th for ture from string 7, tuny a 6s Eb mode. mode, iSartu, dim. Sth DG order to obtain the chosen s — e 7s algorithm is used a second time in — t ht 2 kitmu, 4th = ec tuning be obtained with depar- EC Bb e A t initial gablitu, augm. 4th À Ss t 9 — E isartu, Sth E 8 137 t so that the dichord nid gabli becomes clear, | and so on. The way to roceed is shown clear] by the 7/3 star diagram In Fig. 10.4. Now, consider instead —— = Ss Ss Ts 58 7s pitu, 4th —@——@—@É@c Ss 68 t t s s = t erved t end irst I a he one of Lust IN- 4 embubu, dim. Sth ritnm, e one 9 VII 74+, $ 2, which can G $ C F Bb y Eb Ab + Db y Ts Ss F Bb Ss Fb Ab 7s de Ts Ss Ts F Bb ir e Eb Ss Te Db — Ss Fe Ts bb — n nid ba tos $ 2.5. t 5s Ts Ss St Gb CC a Bb Ab Gb F Eb Db - Ss co bb HE === 1 68 | t t t = + t so stai mode Cb y F Cb Bb Ab Gb F Eb Db pa? = | os t t t o third string. The embuhu dichord becomes consot nant (‘clear’). cb t bb À = string instruaablitu mode kitmu dichord is disso. nant, tighten the fourth s rear (= the sixth) string. t The kitmu dichord I D A b If the ment is tuned to the <p embubu mode, and the Pe pa Cb Bb Ab Gb Fb Eb Db cb bb N D and the embúbu ba § 2.6. ye Ss instru- Db AAA SZ: If the string ment is tuned to the pitu ms| T I H Ts 68 s un A gablitu, augm. 4t t | t I beta s t t n 1) S isartu mode (loosened) _ be- Comes consonant. § 2.7. If the string instrument . is tuned to the kitmu mode, and the isartu dichord nant, is dissotighten the samussu (= second) and f = tone, $ = semitone Fig. 10.6. . mode, Gb 53 be explained vaguely as gabli dichord is dissonant (unclear’), tighten the fi Pre N iSartu, c ni§ gabari, dim. 5th I = Db + Bb Eb Ab Db Gb Cb Fb Bb = Eb toos DAL gablitu, 4th ui F pitu, augm. 4th 7s Ab ve as G nid gabli, augm. 4th e hi T = ni3 gabari, Sth A Ab 68 PS Y Bb 3 Dal 78 nid gabli, 4th € C IT ES 5s G I the rear (= ninth) string. A modern (anachronistic) interpretation of The isartu dichord the OB retuning algorithm in UET VII 74+, $ 3. becomes consonant. Tuning by tightening. and ascending fourths, as illustrated by the 7/3 star This first modal retuning algorithm, too, can be figure in Fig. 10.4, will automatically lead to what in explained in terms of the 7/3 star diagram, as in Fig. modern terminology may be 10.7 below. called 7 different descending diatonic heptatonic modes. All the modes in Fig. 10.7 can be obtained as Note that all the modes in Fig. 10.5 can be obtained follows, without the use of the modal retuning algoalso without the use of the modal retuning algorithm, rithm: first the initial tuning algorithm is used in order namely as follows: First the initial tuning algorithm is to obtain used in order to obtain the isartu mode. Then the algorithm is used a second time in order to obtain the the isartu mode. Then the initial tuning

gablitu mode nis tuhri mode nid gabli mode CBS 1766. A Clue to the Provenance and the Date of the Clay Tablet It seems to be clear now that CBS 1766 is a text with a mixed topic. On one hand, there is the geometric topic of three kinds of 7-sided figures, both the 7/3 star figure which is explicitly depicted, the ‘7-side’ (regular heptagon) whose sides are parallel to the sides of the 7/3 star figure, and the 7/2 star figure whose sides are also parallel with the sides of the 7side. Indeed, according to the interpretation suggested in Fig. Fig. 10.7. The retuning algorithm in UET VII 74+, & 2, in terms of the 7/3 star diagram. (T = tightened.) 6.2 above, the text of the partially preserved headings above chosen mode. The kitmu mode, for instance, can be columns 11-iv seems to refer to two of these three kinds obtained with departure from string 6, by first tightenof 7-sided figures. Regrettably, the text of the correing string 6, then tuning string 3 so that the kitmu sponding heading above columns v-vi is not preserved. dichord becomes clear, tuning string 7 so that the On the other hand, CBS 1766 also concerns the embubu dichord becomes clear, and so on. The way to topic of Old Babylonian music theory, made obvious proceed is again shown clearly by the 7/3 star diagram through the labelling of the seven points of the star in Fig. 10.4. In modern notations, figure by the names of seven strings of the sammü, and the retuning algorithm of UET VII 74+, § 2 can be expressed as: through the listing in column 11 of the seven dichords in the order of the sides of the 7/3 star figure. The ordering of the dichords in the order of the In this connection, it is potentially important that sides of the 7/3 star figure, beginning with the isartu traces are preserved also of the inscription in column dichord, seems to have been the prevailing standard. xi, the last column of the table on CBS 1766, close to At least, this is what is suggested by the Assur text the right edge.” The heading over that column appears (VAT 10101) a long catalog of vocal and instrumental to be mu.[bi.1m] ‘its name,’ while traces of the inscripmusic, tions in lines 1 and 2 of the same column can be read where in particular (see Kilmer, Festschrift Landsberger, 267) a list of love songs 1s summarized as in the following way (ll. 45-52): dichords in lines 1-2 of col. ii. 23 iratu Sa e-Sir-te akkadi 23 love songs in the isartu i-[Sar-tum] and K[i-it-mu-um], the names of the If the suggested readings of the preserved traces of mode, Akkadian inscriptions 17 iratu Sa ki-it-me 17 love songs in the kitmu correct, that means that the table on CBS 1766 in a 24 iratu sa eb-bu-be 24 love songs in the embubu mode mode 4 iratu Sa pi-i-te [...] iratu sa ni-id murub, 4 love songs in the pitu mode [...] love songs in the nid [...] love songs in the nis tuhri mode [...] iratu sa murub,-te [...] love songs in the gablitu mode [...] akkadi* [Total ... love songs], Akkadian The same ordering of the dichords can be observed in column 11 of the table on CBS 1766 (see Fig. 6.1 above). the last column on CBS 1766 are certain sense is a close parallel to the table on the famous Old Babylonian mathematical table text Plimpton 322 (Friberg, MSCT 1, App. 8). This observation, in its turn, is important because it means that conclusions can be drawn about both the date and the gabli mode [...] iratu Sa ni-is tuh-ri in provenance of CBS 1766. Indeed, on p. 33 of an interesting paper about “tables and tabular formatting” in cuneiform texts (in Campbell-Kelly et al., The History of Mathematical Tables), Robson writes that “*... there is only one known mathematical cuneiform tablet which is conspicuously indebted to administrative practise. Plimpton 322 has achieved such an iconic status as the Mesopotamian mathematical tablet par excellence that it comes as quite a shock to 3) Collated by G. Frame, personal communication.

139 2 6 3 7 4 1 5 Old B E A D G === C FB A “== ° 1 Ss Ts 2 3 C B 1 R A G F E D eb ===:== zz. 4 5 6 Fi B (5 ut OH ¢ 7 8 + 5s Ss iSartu, Sth 7s 68 s t t t t 9 + t (and a + E A h 1 J Ts a 5s DG C FR T + } + + 7s Ss gablitu, 4th mode C BA met problem 6s G FED eb — ni — = Le al S Î a t t —— S t other) the statement S erite tratt Ss most Babylonian mathemati| cal texts, the verbs in n oy : gablitu, augm. 4th y Ss 1 E 7s 0 2 (roughly corresponding to the English n qablitu past tense), ey” mode verbs in t of the are in the pret- H S the procedure ni$ gabari, dim. Sth while . the . solution are in the durative (roughly corre. sponding to the English . : future . . imperative. A ° tense), Ritter or in further the states that this kind of rigidity , Ef A# D# GH CH F# B E# C# n B m A# ma G# FH Ef D# c# b 2 on of ade served syntax can in be only obthree other genres of Old Bab5s poo FE Ss $8 7s kitmu, 4th Ss 68 n t Miner S t t t t t . ylonian texts, those of iSartu, dim. Sth . . divination, medicine, and jurisprudence. BR E# A# D# GH CH bh ] -———— 8 7s F# mm] a mn TE ARES Ss Ss Ss B# CH BH nt LS 7s OS Y 6s A# G# F# Ef DH c# bit 7, Ss 75 t 7 one kind of divination n ===. isartu mode N Cc (tightened) t In “o; text, fol instance, the . form that oil takes when poured on water is dela iSartu, Sth Fig. 10. 8. gablitu, augm. 4th scribed with verbs in either the preterite or the A modern (anachronistic) interpretation of stative the OB retuning algorithm in UET VII 74+, § 2. (describing a constant state), while the realize how odd it is. Its fame derives from its mathecorresponding prediction is expressed with verbs in matical content: fifteen rows of four extant columns the durative or the stative. In a cited example, the text containing sophisticated data relating to Pythagoras’ theorem. The fact that this data is laid out in a landscape-oriented headed table, with a final heading MU.BI.IM (‘its name’) for the non-numerical data, has gone completely unremarked. These, of course, are formal features of administrative tables from Larsa during the period of rigorous standardization in the 1790-80s BCE.” Just like on Plimpton 322, the data on CBS 1766 are laid out in a landscape-oriented table with headings, in particular with a final heading mu.bi.im for the non-numerical data. Therefore, the conclusion must be that CBS 1766, like Plimpton 322, in all probability is an Old Babylonian text from Larsa, dating to the period 1790-1780 BC. says If, from the middle of the oil, two drops came out and one was large and the other small, the man’s wife will give birth to a boy; for the sick man: he will recover. In the case of medical procedure texts, the presentation of the medical problem is expressed with verbs in the preterite or the stative, while the medical solution to the problem is expressed with verbs in the durative. In a cited example, the text says If a man was stung by a scorpion, you will apply ‘ox excrement’ and he will recover. In juridical procedure texts, finally, the presentation of the case 1s expressed in terms of verbs in the preterite, followed by a verb in the perfect (English present perfect), while in the solution to the case the UET VII 74+ in the Context of Old Babylonian “Rational Practice Texts” verbs are in the durative. In a cited example, the text says If a man accused a(nother) man and charged murder (against In a section of Ritter’s interesting paper in Chemla, History of Science, 177-200, a typical Old Babylonian mathematical text 1s considered in the context of what him), but he was not convicted, his accuser will be killed. Note that a conspicuous (and typical) common is called there “rational practice texts.” Ritter examfeature of the three cited examples is that they all start ines the grammatical structure of the text, and specifiwith the word cally the verbal chains. What he finds is that in this mathematical texts, on the other hand, rarely start this (Akk. summa). Old Babylonian

way, although they do in a few instances, such as the 5 gur geometric algorithm text VAT 8393 (Friberg, Amazing 1 bal [gür] Traces, 434), and the Eshnunna texts IM 52301 (Hoyrup, Lengths, Widths, Surfaces, 213) and IM 67118 5 (for the) circle 1 (for the) ratio [of a circle] 20' dal [gür] 20 (for the) transversal [of a circle] refer to the following well known Babylonian rules for the area A and diameter d of a circle: (= Db,-146) (ibid., 257). A=5 (: 1/60) = 1/12 This conspicuous feature is shared also by each fora circle with a circumference of unit length paragraph in the retuning algorithm text UET VII 74+, d = 20 ( 1/60) = 1/3 §§ 2-3. Moreover, in each one of those paragraphs, the for a circle with a circumference of unit length. statement of the problem with the given tuning of the sammú instrument is expressed with verbs in the síative, More specifically, this means that (approximately) A = S ( while the solution to the problem is expressed in terms 1/60) : square of a for a circle with the circumference a of verbs in the durative. Therefore, it appears that the d = 20 (- 1/60) = 1/3 : a for a circle with the cirretuning algorithm text UET VII 74+, §§ 2-3 belongs cumference u. to the same category of “rational practice texts” as Old Babylonian mathematical procedure texts, divination texts, medical procedure texts, and juridical pro- The mention of the ‘ratio’ 1 is without known precedent. Presumably, it refers to the ratio a/1 which, of course, is equal to 1 in a circle with a circumference cedure texts! of unit length. As shown in Figs. 11.2-4 below, the text on the reverse of CBS 10996 contains fairly well preserved 11. CBS 10996. A Neo-Babylonian(?) Table of Constants parts of three columns of text, presumably columns ivvi. The general layout of the text is shown in the outline of the fragment below. 10996, a The table of constants on CBS 10996 has a very large fragment of a Neo-Babylonian(?) table of coninhomogeneous, mixed content, just like a number of Photos of obverse and reverse of CBS stants, were published by Kilmer in Or 29, together other known Old Babylonian tables of constants (sce with a translation of the text and a commentary. The Friberg, in Changing Views, 64-67; Robson, Mesoponew copies of the text in Figs. 11.1 and 11.4 were tamian Mathematics, xii, 193-207). The explanation is probably that there existed no “canonical” table of kindly made for the author by F. Al-Rawi. The text on one side of the fragment is almost constants. Instead, tables of constants may typically perfectly preserved, while very little remains of the have been produced by teachers of mathematics who text on the other side. Although the imperfect state of sporadically made notes, for future use in the classpreservation of the clay tablet makes it difficult to be room, of numerical data that they found in mathematiabsolutely sure, apparently the well preserved side of cal (and other) texts that happened to be available to the tablet is the reverse.* them. Furthermore, entries in Babylonian tables of Only 15 lines of the first column on the obverse are constants are usually so brief that it is impossible to partially preserved, but the appearance of, for instance, understand what they refer to, except in the lucky the terms sag ‘front, short side,’ us ‘length, long side,” cases when texts are known where the constants apdal ‘transversal,’ and gur ‘curve, circle’ in this brief pear in a comprehensible context. list makes ıt obvious that this is what remains of a Thus, for instance, the first preserved section in table of constants with parameters for simple plane col. iv on the reverse of CBS 10996 contains a list of geometric figures. In particular, the lines (1 9'-11') constants in some way related to heaps of Se.g18.i ‘sesame.’ Since no text, or at least 1 no obv. mathematical text, is known Where such constants appear in a | 3a Le} 7 3 2 o — ce 3° 64 ‘ | | natural way, there 1s no obvious | 4° | sag x explanation for these sesame con- - Sag] 7 stants.? TS — b t EYN>») Yx *) Collated by G. Frame, personal x communication. = °) Some Neo-Assyrian tablets from an archive in the South Palace of Nebuchadnezzar Il in Babylon contain Hand copy: F. Al-Rawi accounts mentioning both sesame and sesame oil, with an exchange rate of |] unit of oil for 6 or 7 units of sesame Fig. 11.1. CBS 10996, obv. Conform transliteration and copy. Copy: F. Al-Rawi. Pedersén, Studia Orientalia

The second section nn - mm mm ee mn un mn un ue ue un ue ue ue eee ue ue un ue nn un ue du in col. iv, on the other hand, contains constants parameters for well known from astrosesame nomical texts, such as for bricks ee the astronomical and asloading numbers(?) parameters for for straw trological compendium astronomical m parameters for Enuma Anu Ellil (see problems plane geometric Friberg et al., BaM 21, string names and figures parameters for dichord numbers 496-499 pomegranates and Robson, Mesopotamian irrigation problems carrying numbers(?) for straw, etc. — ne nn mn me ms me ue un un un ue un ne ue ue ue ue ne un un ne ue ne c. 38 lines un ue un un un ee _0—_<-- c. 38 lines _ ematics, u. parameters for Sec. Math8.2). In particular, mü and Sú Sa “30 mean the “first rising’ and the “first setting’ of the moon (‘the god 30’), means igi.du,.a ‘visibility’ (of en c. 38 lines the moon), while u,-mu Fig. 11.2. CBS 10996. An outline of the fragment, with and Be Mean day” and an indication of the general layout of the text. night. 6 vi pon ee v tuh-vi ni$ ru $e- esa sa 26 sa i- Sar- Mg | 16 sa sal- Sa- IU y i i 3 7 2,9 sa sa em- buud | | bu NOLI sa sub. 5 7 sa Li muruby- {us er kitsa pisa en 7 3 sat 7 4 4 6 igi. gub. e | zi- 77 ‘ 3 sa 3' ub-ri mu —— 3 ki > 12° ki. 2 er igi. gub. e se. gif. lpg bari gis. ba. ri. ga gis. ba. Ln ga Sn 2» gi5.bán ri. ga ba.- ri. gi. lugal - sa qud-mu-ü à sa4uh-ri 16 Sa Sal-5a-u 4)", ù sa 3-50 uhri 37'sa em-bu-pu i ci x = 4-tu sa 3-5u Si 5 Pau sig 1 3 sa gu a ü sa sa Eb: ges 5 2 sa murubs! Si 8% doth Das tur mur 63sa kicmu sa %- ge US se z sa 4 uh-ri ü SA 3-4 sig à sa“ uh- ri = dal E dal gis. N. ga ti dal gis. ri € di «ri. ga ga gis.bin a . kg a lu ames fu, A PP= À ur. ma_Si-na- nu . Ü nu. nu . ur. ma Si-na- nu ty di ud-ar + == ame ame gis. zi- ri- qum ti patti pat- Ita 18 || 3 a.Sà a.mes Sa- qu- ú ki2 2bn BA 464° a. 8a ame’ ja- qu-ú ki. 2 sin gis kakHiS. : ra iTbg ha-a- mu gún gismar. gid. da . ir whine sia Di) | 3 45 zabalu2464 x gún i ci 12 3 zabadu I, in mu gin po pi i gis. à. 1a a.sàa. mes sa qu u Se gun | fi Tu mi 939 ame lugal I i i 3° pala 11°2 148 àm = i ames Ipgbarig A sa Sünder è sn 14 6 sa zir-d | ca dé, a. di 4 ja ge, a-na uy-mu na-pa- 36 bg 2bn dal giS.ri. ga sa 33ú sig u sa en =Mr i a a = L gur dg Zon] dal gis. ri. ga dal gis.ri. ga i i dus. 4 Ls E5 57 xlgin:2 2 gin :6 x- Sun 1 gs gis. 33° gii. — nu„452 2_ gin 51'Sgin: 4'l'gin:3 4 5 sa pe | gi le 1 sùupmua-na ge,Ja na- ©pa- lu3 è sa Sá- ge, usa 3-5 uh-ri 27' sa Cia gis. ban TT, ila muti 4 1 sa Sub. mudado. 3 a. 12 2sila : Y 11 s : — b.be dé. a. dí ù sa qud sila T 3 3' sila: 64 7'sila: 6 5 i =[2 nl “Tog 2bn ais. baba. ri. ga usa 5-511 75 sa see ‘hl - lg Delia gur igi gub. € n ! gis. >2 = + = sigg. 3- fi i Se. ema Ta ginigsa. hà Sa- ge, USI 4 ub-ri 2 6 sa ¿Sarita | — sa ragip 7 ANA 3 me Sig4- ali.= ig A sa qud-mu-t à sa 5-5u 15 sa nis sa UE: ki. 2 gi. Su. kin | A. 73 Se. Ta gumigsaba bash 13 4 5 ig a si tr Sar- tug | | e mia 1°2 HE mé. 1a Ge im-ma- ti 3 ST i | gis. Sub.ba sa | | € DI | muruby | | 4 1 1 3 - rev. PP ee ee ee ae eee en ee ee ee EEE x 1 5 75 i iv > za-ba-lu lbg barig i 1 \ ' I — — — —— — — " cocco + + one ue mu + ue ne n o ne en on en on on on en en ee ee en mn ee on ee mn one on mn ee me + en ee en = ee nn m m m m m m m m m = = ee ee ı Fig. 11.3. CBS 10996. A Neo-Babylonian table of constants of mixed content. Conform transliteration. 198. The constants for sesame mentioned in CBS 10996 are discussion of the oil pressers on pp. 261-287. Various rates of a different nature. See also Bongenaar, Ebabbar, with a are mentioned in fn. 241 on p. 266.

A 7 È PT: Fig. 11.4. CBS 10996. A Neo-Babylonian table of constants of mixed content. Copy F. Al-Rawi. concerned with similar way, with constants called giS.ma.la, only this constants for giS.nu.ür.ma ‘pomegranates.’ In this case, The third section in col. iv, is time not for bricks but for reeds and reed bundles. The too, the term does not appear in any known matheexact meaning of these reed constants is not known. matical texts, so the meaning of the constants remains unknown. On CBS 10996, the table of constants in the proper sense ends with the section of reed constants. What The fourth section is concerned with constants for then follows, in the lower part of col. v, is not a table irrigation (a.meS means ‘water’). Several of these conof constants but a seemingly peculiar enumeration of stants are known from Old Babylonian mathematical sexagesimal numbers and capacity measures, making texts (see Robson, Mesopotamian Mathematics, Sec. no real sense in a table of constants. It is rather in 6.5). several ways similar to the table of parameters for a The fifth and final section in col. iv contains conseries of mathematical problems concerned with measstants in some way associated with problems concernuring vessels in the Old Babylonian theme text YBC ing transportation of commodities measured in capaci- 4669. (See Friberg, MSCT 1, Sec. 4.7, in particular ty measure. In particular, gún giS.mar.gid.da means Fig. ‘load of a wagon,’ and zabalu means ‘to carry.’ discussion in Friberg, BaM 28, 309) that the numbers In col. v, the first preserved section mentions constants called giS.ma.la ‘cargo-boat,’ meaning either “molding numbers” or “loading numbers,” for four kinds of bricks, namely sig, bricks, measuring 1/2 cubit x 4.7.) It is likely, therefore (see the clarifying and capacity measures tabulated in the lower part of col. v on CBS 10996 are the data for two series of exercises much like the ones in YBC 4669, one series (ordinary rectangular for box-like measuring vessels, and a second series for 1/3 cubit x 5 fingers), cylindrical measuring vessels. sig,.ab (half-bricks, 2/3 cubit x 1/3 cubit x 5 fingers), In view of the proposed explanation of mathematisig,.al.ur.ra (square bricks, 2/3 cubit x 2/3 cubit x 5 cal fingers), or sig,.2/3-ti (larger rectangular bricks, 18 work notes for future use in the class room, there is 12 fingers * 5 fingers). Such constants are nothing strange in the inclusion in this text of data for fingers x well known from various Old Babylonian problem texts and tables of constants (see Friberg, in Changing Views, Sec. 4.1; MSCT 1, Sec. 7.3). The second section in col. v is structured in a tables of constants as a mathematics teachers’ a couple of series of mathematical exercises. Similarly, there is nothing strange in the inclusion of the table of names for fourteen dichords in the upper part of col. vi of CBS 10996, actually at the

very end of the text, after the author of the text had run The badly preserved text out of mathematical constants that he wanted to make contains a large number of entries, but most of those G = IM 49949, admittedly notes of. Also, it is not strange that when he saw that entries list mathematical problem types, not constants.) there was still some space available in the lower part Against this background, it is most unfortunate that so of col. v, he decided to be more explicit, calling by much of the obverse of CBS 10996 is lost, but so name not only the fourteen dichords but also the seven much more fortunate that column vi with its musical strings in terms of which the dichords were defined. terms 1s so well preserved! It is more surprising that the primary series of dichords is not recorded in the same order as the sides of the 7/3 star figure (2, 6; 6, 3; efc.; see Fig. 10.4), but rather in lexicographic order (1, 5; 2, 6; etc.). The reason may be that the author of the text did not really understand the tuning procedure based on the sequence 2, 6; 6, 3; efc. The order of the secondary series of dichords (7, 5; 1, 6; efc.) seems to be coupled (in a somewhat unorganized way) to the order of the primary series of dichords. The purpose of this secondary series of dichords is not well understood. It has been proposed, in Smith and Kilmer, SMA I, that the secondary series of dichords was used for “fine tuning,” but according to Dumbrill (Archaeomusicology) the proposal is unre- 12. On Greek “Pythagorean” Music Theory and Ratios of String Lengths It would be extremely difficult to try to give a brief and comprehensible account of ancient Greek music theory in general, for a number of reasons. The account below will be concentrated to a narrow and limited discussion of only sources for what is known about Greek music theory in the particular cases when it is concerned with diatonic heptatonic scales of the Babylonian type, mainly expressed in terms of ratios of string lengths. alistic. So, maybe, this secondary series was contrived, in a purely theoretical way, as an attempt to give a Pythagoras as the Alleged Discoverer of Epimoric musical meaning to the dichords corresponding to the String Ratios sides of a 7/2 star figure, as suggested by the (incomplete) evidence of CBS 1766 (Fig. 10.2). The discovery that musical consonance is directly related to numerically simple ratios of string lengths It is remarkable that Babylonian music theory seems was attributed to Pythagoras himself by his followers, to have been closely connected with Babylonian mathethe so called Pythagoreans. How the discovery allegmatics. This is shown not only by CBS 10996, where edly was made is described in a well known, but the names of the 14 dichords are recorded in a mathecertainly matical table of constants, but also by CBS anecdote, which begins as follows: 1766, both historically and physically incorrect where three kinds of 7-pointed star figures (a regular heptagon, a 7/2 star figure, and a 7/3 star figure) apparently are considered both as geometric objects and as a visualization of the 14 dichords. Last, but not least, the retuning algorithm in UET VII 74+, col. ii is Nicomachus, Enchiridion, Ch. 6 (the beginning of the 2nd century BC; Barker, GMW II, 256-258) “... Happening by some heaven-sent chance to walk by a black-smith’s workshop, he (Pythagoras) heard the hammers beating iron on the anvil and giving out both in form and in context very much reminiscent of sounds fully concordant in combination with one ana mathematical recursive algorithm. (Cf., for instance, other ... and he recognized among them the consothe ascending and descending geometric recursive alnance of the octave and those of the fifth and the gorithms in VAT 8393, Friberg, Amazing Traces, App. fourth. He noticed that what lay in between the fourth 1, used for the construction of a chain of trapezoids with fixed diagonals.) and the fifth was itself discordant, but was essential in filling out the greater of these intervals ...’ Note: There seem to have been about 38 lines in each column on the reverse of CBS 10996, and it is warranted to assume that also the three (or four?) columns on the destroyed obverse contained about 38 Pythagorean String Ratios in Plato’s Timaeus One of the oldest known references to ratios of lines string lengths is contained (implicitly) in a famous each. Since the table of constants in the proper sense passage in Plato’s dialogue Timaeus, which describes ends in the middle of column v, a reasonable estimate how the divine ‘Craftsman’ began his creation of the is Soul of the Universe by dividing a mixture of the that CBS 10996 originally something like 4 : 38 + 14 = mentioned, at least, 166 constants. This makes CBS 10996 (in its original, intact form) by far the most extensive of all known Babylonian tables of mathematical or technical constants. (Compare with, for instance, TMS 3 with 70 entries and YBC 5022, Neugebauer und Sachs, MCT text Ud, with 66 entries. Same, the Different, and the Being into a number of components in the following way: Plato, Timaeus [35b-36b] (the first half of the 4th century BC; Barker, GMW II, 59-60) “... This is how he began to divide. First he took away one part from the whole; then another, double the size

of the first, then a third, hemiolic with respect to the mean between two given numbers (integers) p and q second and triple the first, then a fourth, double the can be computed as follows: second, then a fifth, three times the third, then a sixth, ifp+r:p=q-rgq for some part r, then p-g=reight times the first, then a seventh, twenty-seven (p + q), and so on. times the first.” “Next he filled out the double and triple intervals, once again cutting off parts from the material and The other kind of mean between two numbers p and gq can be computed as follows: placing them in the intervening gaps, so that in each ifp +n =q-—n interval there were two means, the one exceeding and SO exceeded by the same part of the extremes themselves, the other exceeding and exceeded by an equal number. From these links within the previous interfor some number n, then p — g = 2 n, and On. In Nicomachus’ Enchiridion or ‘Handbook’ of harmonics (2nd century AD), Ch. 8 (Barker, GMW II, vals there arose hemiolic, epitritic and epogdoic inter- 259), the following example is used to clarify the vals; and he filled up all the epitritics with the epogdoic situation: kind of interval, leaving a part of each of them, where ““... A duple interval is that of 12 to 6; and it has two the interval of the remaining part had as its boundameans, the numbers 9 and 8. Now the number 8 is a ries, number to number, 256 to 243. And in this way mean in harmonic proportion between 6 and 12, exhe had now used up all the mixture from which he cut ceeding 6 by one third of that 6, and exceeded by 12 these portions.” by one third of that 12. ... The other mean, which is Strange terms in this passage, borrowed from Py- 9, and which is so placed as to correspond to parumese, thagorean music theory, are ‘hemiolic,’ from Greek is reckoned to stand as an arithmetical mean in relahemiolios (‘a half and a whole,’ tion to the extremes, exceeding 6 by the same number, meaning 1 1/2), ‘epitritic,’ from Greek epitritos (‘a third more,’ meaning 1 1/3), and ‘epogdoic,’ from Greek epögdoos (‘an eighth more,’ meaning 1 1/8). These three expressions are all more,’ ‘epimoric,’ from Greek epimorios (‘a part meaning 1/n, 1 3, as that by which it is exceeded by 12. ...” The example is well chosen, because it is easy to see that 12=2:-6,8 12=1 =11/3-6,9=11/2:.6,9=1 1/8 - 8, 1/2: 8, and 12 = 1 1/3: 9. for some small integer n). Incidentally, such expressions are not unusual in Old Every one of these relations is intimately connected Babylonian mathematical texts, where they typically with Pythagorean music theory. occur as coefficients in quadratic equations. See, for instance, §5 of the Old Babylonian mathematical The Euclidean Division of the Canon catalog text BM 80209 (Friberg, Amazing Traces, 29). Also the meaning of the remainder of the cited Note that commonly used translations of the kind paragraphs from the Timaeus will become clear after (n + 1)/n for ‘epimoric,’ and 3/2, 4/3, 9/8 for “hemiolic,” the continued discussion below of important examples ‘epitritic,’ ‘epogdoic,’ are somewhat anachronistic. of Pythagorean music theory, beginning with selected Indeed, the earliest documented use of common fracpropositions from the little treatise Sectio tions occurs (implicitly) in the Egyptian demotic mathe- ‘Division of the Canon’, which is attributed to Euclid matical papyrus P.BM 10520 §5 (early(?) Roman). in several of the known sources. (See Friberg Unexpected Links, 150-155.) Thus, when Plato awkwardly writes “the remaining part had as its boundaries, number to number, 256 to 243,” that is The Euclidean Canonis (‘Division of the Canon’) (Barker, GMW II, 190-208) Prop. 6. The duple interval is composed of the two precisely because he can only express as a ratio (less greatest anachronistically, 256 : 243) what we would write simepitritic. ply as the common fraction 256/243. Sectio Canonis epimoric Two proofs intervals, of this the hemiolic proposition and the are given. The In the first of the cited paragraphs from the Timaeus, second, simpler proof argues as follows: If A is the Plato divides the divine mixture into parts of the hemiolic of B and B the epitritic of C, then A contains relative sizes B and half of B. Therefore two A’s are equal to three a, 2 a, 1 1/2 : 24,2: 24,3: (1 1/2 - 2 a), 8 a, 27 a. These relative sizes can also be expressed as la, 2a,3a, 4 a, 9 a, 8 a, 27 a, where 4 and 9, 8 and 27 are the squares and cubes, respectively, of 2 and 3. B’s. Also B contains C and a third of C, so that three B’s are equal to four C’s. Therefore, two A’s are equal to four C’s, so that A is equal to two C’s. Hence A is double G.° Prop. 8. If an epitritic interval is subtracted from a hemiolic interval, the remainder is epogdoic. In the second of the cited paragraphs, Plato sug- In the proof, it is assumed that A is the hemiolic of gests that epimorics can be inserted between the pow- B and C the epitritic of B. Then A contains B and half ers of 2 and 3 (“the double and triple intervals’) by use of two kinds of means, namely what we would call the harmonic and arithmetic means. The idea is that one 5) Anachronistically, in terms of common fractions: if A B and B = 4/3 C, then A = 32 - 4/3

of B, so that eight A’s are equal to twelve B’s. Again, fourths and a tone, so that two fourths are less than C contains B and a third of B so that nine G’s are five tones. And so on. equal to twelve B’s. Consequently, eight A’s are equal Prop. 19. To mark out the canon according to the socalled changeless system. to nine C’s. Therefore A is equal to C and an eighth In the ‘changeless system’ of Greek music theory, a of C. Hence A is the epogdoic of C.’ central octave is thought of as composed of two Prop. 9. Six epogdoic intervals are greater than one duple interval. ‘tetrachords’ (four successive strings), separated by a In the proof, it is assumed that A is a number, that tone. In Fig. 12.1 below, the two tetrachords of the B is the epogdoic of A, C of B, D of G, E of D, F of central octave are called the ‘middle’ and the ‘disjoined’ E, and G of F, so that A, B, C, D, E, F, G are tetrachord. To the central octave are joined on either epogdoics of one another (a geometric progression). side an ‘extra’ and an ‘upper’ tetrachord, and then an Then, in the least terms (see Euclid’s Elements VIII additional tone. The result is strings spanning a double 2), octave. The ‘canon’ in Prop. 19 is a measuring stick (a A = 26 myriads 2,144 (the sixth power of 8), “monochord”) along which a string is stretched from B = A + 1/8 A = 29 myriads 4,912, C = B + 1/8 B = 33 myriads 1,776, A to B (see again Fig. 12.1, which, by the way, is D = C + 1/8 C = 37 myriads 3,248, more detailed than the corresponding diagrams in the E = D + 1/8 D = 41 myriads 9,904, original manuscripts). F = E + 1/8 E = 47 myriads 2,392, A moveable bridge can take any position from E to A, defining a corresponding string G=F+ 1/8 F = 53 myriads 1,441 (the sixth power of 9). length with departure from B. The positions of the Hence G = 53 myriads 1,441 is more than two A’s bridge producing notes corresponding to various parts = 52 myriads 4,288.8 of the ‘changeless system’ are determined algorithmi- Prop. 12. The octave interval is duple. cally, in a sequence of steps, in the following way: In the proof, proceeding in a not quite satisfactory 1. The bass note is defined by the whole string AB. It is way from the axiomatic assumption that concordant called proslambanómenos, the ‘added-on,’ intervals correspond to string ratios that are either multiple or epimoric, 2. AB is divided into four equal parts, at C, D, and E. it is observed, among other Then AB is epitritic of AC, so that CB is a fourth things, that the octave is made up of the hemiolic and above AB in pitch. It is called the ‘upper diatonic.’ the epitritic, the two largest epimoric intervals. It is also made up of the fifth and the fourth, A —_— A" added-on and these are both o epimoric (Prop. 11). Therefore, the fifth E is hemiolic, the fourth is epitritic, and upper hypäte L = the octave is duple. L > ST Prop. 13. It remains to show that the > E interval of a tone is epogdoic. a Indeed, if an epitritic interval is (upper diatonic) subtracted from a hemiolic interval, the . remainder is epogdoic, and if a fourth Therefore, the interval of a tone is epogdoic. Prop. 14. The octave 1s less than six PHO) YE > H 2 S ze = "SC next to mése o E TD o E a E Pas This follows from Prop. 9. 2° disjoined néte ¿2-7 G extra nete 2 È E LAR H- E-- 7) Anachronistically: if A = 3/2 B and C = 48 7 Vs |” J a TA “|b F-yve than three and a half tones. is equal to two Tr K LE © | conjoined néte . o This is because the octave, which 1s 1.71 2 tones. Prop. 15. The fourth 1s less than two and a half tones, and the fifth is less 9 3 [5° —P | 23 o 5 middle hypaie 357 a Q | is taken from a fifth, the remainder is (by definition) a tone. But the fifth is hemiolic and the fourth is epitritic. less than six tones _ y i a ; proslambanómenos *added-on hypatos ‘upper’ (rel. to the monochord) hypáte “top (relative to the monochord, not in pitch) diátonos ‘diatonic’ parhypate ‘next to hypate’ mésos ‘middle’ mése “middle” diezéugmenos ‘disjoined’ = 4/3 B, then B = 3/4 C and A = 3/2 - 3/4 paramése ‘next to mése’ C = 9/8 C. néte ‘bottom’(rel. to the monochord) B synémmenos ‘conjoined’ hyperbolaios ‘extra’ $) A myriad is 100 times 100. Anachronistically: G = (9/8) - A = 531,441/262,144 A is more than 2 A. Fig. 12.1. Sectio Canonis, Prop. 19. Construction of the fixed notes (independent of genus).

4. OB is made hemiolic of XB. Then OB is a fifth below 3. AB is made duple DB. Then DB is an octave above XB. It is called ‘middle next to hypate.’ AB. It is called mése, ‘middle.’ 5. PO is made equal to OX. Then PB is duple XB, so 4. AB is quadruple EB. Then EB is two octaves above that PB is an octave below XB. AB. It is called ‘extra néte’ (néte = ‘bottom’). It is called ‘upper next to hypate.’ 5. CB is made duple FB. Then FB is an octave above 6. CB is made epitritic of RB. Then RB is a fourth CB. It is called ‘conjoined nete.’ above CB. It is called ‘middle diatonic.’ 6. DB is made hemiolic of GB. Then GB is a fifth above The distribution of tones and ‘semitones’ in the double DB. It is called ‘disjoined nete.’ 7. HB is made duple GB. Then HB is an octave below octave, the “Greater Perfect System,” is not explicitly GB. It is called ‘middle hypate’ (hypáte = ‘top’). mentioned by the author of Sectio Canonis, but it is 8. HB is made hemiolic of KB. Then KB is a fifth above easily determined, for instance as follows: HB. It is called ‘next to mese’. 7. By construction, MB is a tone below EB, and NB a 9. LB is made duple KB. Then ZB is an octave below tone below MB. KB. It is called ‘upper hypate.’ 8. GB is a fourth below EB, at the same time as NB is In this way, the string lengths corresponding to all the two tones below EB. If the amount that GB is below ‘fixed notes’ of the ‘changeless system’ have been NB is called a ‘semitone,’ then a fourth can be determined, and also the string length corresponding divided into two tones and a semitone, where, accordto one ‘moveable note’ (CB). The fixed notes bound ing to Sectio Canonis, Prop.15, a semitone is less five tetrachords and two tones of the double octave. than half a tone. In terms of string ratios, GB = 1 1/3 - EB, and NB Note that the procedure used is entirely mathematical. = 1 1/8 - 1 1/8 - EB. Consequently, 3 GB = 4 EB, and The author of Sectio Canonis did not bother to 64 NB = 81 EB, so that 81 - 3 GB = 81-4 EB = 4 demonstrate that the construction really yielded the - 64 NB. In other words, 243 GB = 256 NB, or desired result. It is easy to supply the missing details, GB : NB = 256: for instance as follows: from Plato’s Timaeus.) 10. DB is an octave below EB and a 9. XB is a fourth below NB, and KB is a fourth below fifth below GB. GB. Therefore, KB is a semitone below XB. Therefore, GB is a fourth below EB. 11. GB is an octave above HB and a 10. OB is a fifth below XB, and HB is a fifth below KB. fifth above DB. Therefore, HB is a semitone below OB. Therefore, DB is a fourth above HB. 12. HB is an octave below GB and a 243. (Cf. the cited obscure passage 11. PB is an octave below XB, and LB is an octave below fifth below KB. KB. Therefore, LB is a semitone below PB. Therefore, KB is a fourth below GB. 13. HB is a fifth below KB and a fourth below DB. Therefore, DB added-on = 2 14. KB is an octave above LB and a fifth above HB. Therefore, HB is a fourth above LB. A o SV vies y _ È A 15. AB is an octave below DB, LB is a fourth below HB, and HB is E p 2 Prop. 20. It remains to find the ‘moveable notes’ are the notes defined by seven strings within the four tetrachords. One of these has been found already, the “upper .— middle hypate — middle next to hypate E misa next to mése disjoined trite conjoined néte notes are determined algorithmicaldisjoined néte xtra ly, as follows: fr extra diatonic 1. MB is made epogdoic of EB. Then MB is a tone below EB. It extra néte — © TD & O E © 2 T2 o vi ri = dò so S-— O a8 =” y - K gs "© N < E M pa AE mn 2 5 77) ms O E 3. XB is made epitritic of NB. Then XB is a fourth below NB. It is called ‘disjoined trite.’ x % E + 3 © E È 2 AN 3 = = = o = = J ” © co y = ‘third’). X + is (trite AS un Then NB is a tone below MB. It trite’ AZ : R +» O is called ‘extra diatonic.’ 2. NB is made epogdoic of MB. I ~ Zs middle diatonic diatonic.’ The remaining moveable ‘extra - e men Y 2 E moveable notes. called | ~ cg upper diatonic Therefore, AB is a tone below LB. The | .— upper hypáte upper next to hypate a fourth below DB. re , is a tone below KB. $ © 6 E Fig. 12.2. Sectio Canonis, Prop. 20. Construction of the moveable notes in the diatonic genus.

will be, peculiar to each of them, a specific note of 12. AB is a fourth below CB, and a tone and a semitone the octave that belongs to dynamic mese, since the below PB. Therefore, PB is a tone below CB. 13. LB is a fourth below HB, and a tone and a semitone tonoi are equal in number to the species. For if we set below CB. Therefore, CB is a tone below HB. out an octave in the intermediate range of the com- 14. CB is a fourth below RB, and HB a fourth below DB. plete systéma, that is, the range from thetic middle Therefore, RB is a tone below DB. Then, OB is also hypate to disjoined nete (to allow the voice to move about and exercise itself comfortably upon melodies a tone below RB. of middling compass, for the most part, going out 15. CB is an octave above FB, and four tones and two semitones below XB. infrequently to the extremes because of the hard work Therefore, XB is a tone below FB. Then, FB is also and force involved in slackening or tension that goes a tone below GB. beyond the norm), the dynamic mese of the Mixolydian The distribution of tones and semitones is indicated in will be attuned to the position of the disjoined next to Fig. 12.2 above, which, by the way, is more detailed nete, so that the tonos may make the first species of than the corresponding diagrams in the original manuthe octave in the range set out; that of the Lydian will be attuned to the position of the disjoined trité, correscripts.” sponding to the second species; that of Phrygian to the position of the next to mesé, corresponding to the Ptolemy’s Construction of the Seven tónoi (Octavethird species; that of Dorian to the position of the Forms) mesé, making the fourth and central species of the octave; The distribution of tones and semitones is explicthat of Hypolydian to the position of the middle lichanos, corresponding to the fifth species; itly mentioned in the following interesting passage in that of Hypophrygian to the position of the middle Ptolemy’s Harmonics. next to hypate, corresponding to the sixth species; and that of Hypodorian to the Ptolemy, Harmonics, II.10-11 (Egypt, 2nd century position of the middle hypaté, corresponding to the seventh species. .... AD; Barker, GMW II, 336ff.) In the cited passage, Ptolemy demonstrates a sim- 11.10 “... This (the production of modulations) can be done according to the proper method, if we begin ple procedure by use of which seven modulations of by setting down a higher tonos, which we call A, then an initial tónos can be produced, each one with its take first the one lower than it by a fourth, B, and next mése located within the central octave of the Greater the one lower than B by a fourth, C, which will still Perfect System (GPS), the octave most suitable for the be within the compass of an octave. Next, since the voice. He begins with a “higher” (actually, the highone lower than C by a fourth falls outside the octave, we take the one functionally equivalent to it, that is, the one higher than C by a fifth, D. Then, once again, we set down the one lower by a fourth than this one, est) tonos A, then considers B a fourth “below” A, and C a fourth below B. Since three fourths extend over more than one octave, a fourth tónos cannot be pro- E, and next, instead of the one lower than E by a duced by going down by another fourth. Instead, the fourth, since that too falls outside the octave, we next Zonos, called D, is produced by moving C up by make F the one higher than E by a fifth; and we set a fifth. Similarly, the fifth, sixth, and seventh fonoi, down once again the one lower than F by a fourth, G. called E, F, and G are produced by going down a ... It will unquestionably follow that the differences fourth, up a fifth, and finally down again by a fourth. between C and E, between G and E, between B and D, and between D and F are constituted as tones, while those between G and B and F and A contain what is called the limma. ...” “Now A corresponds to Mixolydian, F to Lydian, D to Phrygian, B to Dorian, G to Hypolydian, E to See Fig. 12.3 below. Reordering the seven octave species from the “highest” to the “lowest,” Ptolemy points out that C is now a tone below E, and that similarly the differences between E and G, between B and D, and between D Hypophrygian, and C to Hypodorian, so that the differences tween them, 1. EC = DC - BC = Sth - 4th = tone beh which have been somehow or < down, L other handed 2. GE= FE — DE = Sth - 4th = tone 3. BG = BC - GC = 4th - ditone = imma 4. BC=DE=FG = AB have now been discovered by reason.” 11.11 | y DB = EC = tone, FD = GE = tone, “It is clear AF = BG = limma that in these tonoi that we have set out there 2) This disposition of tones and semitones (and their counterparts in other genera) is absolutely normal in Greek Fig. 12.3. Harmonics 11.10. Six modulations of a higher tónos. syllus, Nichomachus, Ptolemy, etc. (A. Barker, personal communication.) I want to use this opportunity to thank A. accounts, in Aristoxenus and his followers as well as in Barker for gently guiding me through some of the intricacies exponents of mathematical harmonics such as Plato, Thraof Greek music theory.

dn don - dt - el - nm 7 m dn - dan - di - > nw 7 m - mi + §& enn - dn el - dna dt - nm - enfa-o - uh - enn - enfa-o - An 7 - dnn E mi - = moe > nm 7 È > — mh JJ? - ul unh - uh - en/a-o - enn - enfa-0 enn - Om IE fai m mi - má - Anh ul — mh un À - uh enlao - - uh ef - enn - ul - sidered to be identical. 1 Although Ptolemy does not make this clear, apparently - enn one fonos is a fourth, a tone, . — or a semitone above another 5 one is that the octave form of & ©! |£ = the latter tónos is the same as di - 5 dna - È da 43 the octave-form of the former i © m - n = dr | nm mnh - mh - whos unh - uh - enlao - a dann ma mo , © . . | & (nos, only rotated downwards = " by y aa fourth fourth, a tone, or a semi- Y tone. > 4 o ul - unh - J dn mí - _ dn - - nm en a er E 2 mnh 4 - enn 4 en/ao | 7 30 142 mna ul unk what he means by saying that da > dnn of the double octave are conun A - dr +. Inh. et - se = ul |< us enfa-o - er - - anh mh enn - uh dt nm — In Harmonics II.11, Ptolemi — mo my explains the seven octavemnA - mi - What he has in mind is probforms in terms of the GPS. mh ul — ably something like the schemnh mh - matic diagram in Fig. 12.4 above (which is an elaboraunh - ui - tion of Barker’s diagram in GMW II, 20, but not part of boldface: fixed notes i = lichanös 'forefinger' nn = next to néfe the original manuscript). In the middle Fig. 12.4. Ptolemy’s Harmonics, 11.10-11. The seven tónoi column of this diagram, the fixed notes of the explained in terms of the Greater Perfect System. GPS are shown to extend downwards from e # = ‘extra and F, all are a tone, while G is a limma ‘remainder’ néte,” through d n = ‘disjoined nete,’ n m = ‘next to (a semitone) below B, and F a /imma below A. mese, m = ‘mése, and m A = ‘middle hypate, to Ptolemy then concludes that A, F, D, B, G, E, C u h = ‘upper hypate’ (and a-o = “added-on”). The are related to the seven Greek octave-forms Mixolydian, central octave extends downwards from dr to m A. It Lydian, Phrygian, Dorian, Hypolydian (a fourth below is comprised of the disjoined tetrachord between d n Lydian), Hypophrygian (a fourth below Phrygian), and and n m, the disjunctive tone between n m and m, and Hypodorian (a fourth below Dorian). the middle tetrachord between m and m h. Finally, he considers the mése of each one of the In this middle column of the diagram, the names of seven fonoi. (Note that if the division of the octave the notes into intervals, called the éidos, meaning ‘form’ or position.’ In the other six columns of the diagram, the are thetic, meaning given ‘according to ‘species’ of the octave, is known for a given fonos, names of the notes are dynamic, meaning given ‘acthen the so called “dynamic” mése of that tónos is cording to function.’ (In the middle column, there is always located between the ‘higher disjunctive tone’ actually no difference between thetic and dynamic.) and the tetrachord below it.) He claims that the (dy- The dynamic names of the notes in the first fonos namic) mése of the Mixolydian corresponds to the can be thought of as being produced in the following “position” of the ‘disjoined next to nete,’ while that of way: The central interval remains unchanged (in posithe Lydian corresponds to the position of the ‘disjoined tion), but the GPS is moved upwards as far as possitrite, and so on. (What this means will be explained ble, until below.) Only in the Dorian octave-form is the dynamic thetic m h = ‘middle hypate.’ (The ‘added-on’ note is mése located at the position of the mése. In order to comprehend what is going on in Haru A = ‘upper hypäte’ takes the place of the disregarded.) Since the GPS possible in the first tónos, is moved as high as this is also called the monics II,10-11, it is necessary to understand what a “highest” tónos. In the other six tónoi, the GPS is tónos is and what it means that one fonos is a fourth moved downwards, one step at a time, until it reaches above another tonos. Essentially, a tonos is characterits lowest possible position, with e m = ‘extra nete’ ized by its “octave-form,” the particular ‘form’ (eidos) taking the place of thetic d n = ‘disjoined nete.’ In this in that tónos of the intervals making up an octave process, as pointed out by Ptolemy, the dynamic mése (with repetition of the same form in the second octave of a double octave). The double octave is assumed to be cyclic, in the sense that the highest and lowest notes moves from the position of the moveable note d n n ‘disjoined next to nete’ to the position of the fixed note m h = ‘middle Aypate.’

Note that Ptolemy’s construction in Harmonics II.10-11 is independent of genus. What counts is only ul 94:49 - 1 1/8 = 106540 oc u unh 106540: 1 1/7 = 121:54 exa’ vd’. 149 the positions of the fixed notes making up the bounda- In the exhibited example the computed numbers are ries of the tetrachords and of the disjunctive tones. not, as would have been expected, strictly contained Thus, for instance, the octave-form of Ptolemy’s “higher” tonos A (Mixolydian) is composed, in debetween 60 and 120, the chosen numbers for the boundaries of the fixed central octave. Actually, an scending order, of a tone and two tetrachords. This 1s inspection of all the seven tables in Harmonics 1.15 what Ptolemy reveals that the computed numbers stay between the elsewhere (Harmonics 11.3, Barker, GMW II, 322-323) calls the “first” octave-form. The expected boundaries for all considered genera only in other six octave-forms are obtained from this first one the by rotating the first octave-form downwards, one step Hypodorian (C). These are precisely the cases when ata time. the boundaries of the fixed octave coincide with fixed cases of Mixolydian (A), Dorian (B), and notes of the tonoi. Ptolemy’s Tables of Numbers for the Seven tonoi in Several Familiar Genera Hypothetical Tables of String Ratios for the Seven Harmonics 11.15 (Barker, GMW II, 352-355) con- Babylonian Diatonic Modes tains a series of numerical tables giving an “exposition Nothing corresponding to the Greek identification of the numbers that make up the divisions of the of concordant string pairs with epimoric ratios of familiar genera in the seven tonoi.” It is not very string lengths is known (so far) from any cuneiform difficult to see how the tables were constructed. First, texts. On the other hand, this fact is quite surprising, the numbers (relative string lengths) for the boundaries in view of the enthusiastic calculations with all kinds of the fixed octave are chosen to be 60 for the ‘disjoined of numbers and measures that are so characteristic for nete’ and 2 : 60 = 120 for the ‘middle Aypate.’ Then many kinds of both Sumerian and Babylonian cuneithe corresponding number for the dynamic mése in the form texts. Mixolydian, a tone below the upper boundary of the experiment to try to figure out what Babylonian matheoctave, is 60 : 1 1/8 = 67 1/2. The mése in the Lydian maticians/musicians could have made of the idea of is epimoric string ratios if they had known about it. a semitone below that, etc. Thus, in Ptolemy’s It is, therefore, an interesting thought tables the numbers associated with the dynamic mése In the first of the star diagrams in Fig. 10.7, the one in each one of the seven fonoi are as follows, indefor the isartu mode, the first side of the star defines a pendent of genus. (Note the use of sexagesimal fracdescending fifth from string 2 to string 6, the next side tions, rounded to the first sixtieth in the tables.) 60 - 67 1/2 11/8 =67 1/2 256/243 = 71 1/9 = 67:30 an ascending fourth from string 6 to EEN = (appr.) 71:07 00 È 119 - 11/8 = 80 mí 80 - 11/8 = 90 90 - 256/243 = 94 22/27 = (appr.) 94:49 op string 3, the third side a descending Ci ney dian) Phe elon) (Dorian) fifth from string 3 to string 7, and so on. Suppose now that the (relative) length of string 2 is 1. Since string 6 is reached from string 2 by a descending fifth, its relative length 94 22/27: 11/8 = 106 2/3 = 106:40 oc’ y (Hypolydian) (Hypophrygian) 106 2/3 11/8 = 120 ox (Hypodorian) : @S' 8 is (theoretically) 1-1 1/2 =1 1/2, or The five genera considered in Ptolemy’s tables are 1) simply 3/2. Since string 3 is reached from string 6 by (a mixture of) “tense chromatic” and “tonic diatonic,” an ascending fourth, its relative length is (3/2)/(1 1/3) 2) “soft diatonic” and “tonic diatonic,” 3) “tonic dia- = 3/2 : 3/4 = 9/8, or simply, in modern notation, 32/27. tonic,” 4) “tonic diatonic” and “ditonic diatonic,” 5) And so on. In other words, the ratios of string lengths “tonic diatonic” and “tense diatonic.” In tonic diain the isartu mode can be computed as follows: tonic, for instance, the tetrachords are divided into string intervals with the epimoric ratios | 1/8, 1 1/7, 1 1/27. 2 Here, of course, 1 1/8 * 1 6 1/7 - 1 1/27 = (9/8 - 8/7 - ratio 1= 1] 1 - 3/2 = 3/2 in 3 3/2 - 3/4 = 37/23 Ptolemy’s table 11.2 (Lydian), column 3 (tonic dia- 7 37/23 + 3/2 = 3/24 28/27 = 4/3 =) 1 1/3. Therefore, in particular, tonic), the numbers are constructed as follows, with departure from Lydian mése = 71 1/9: dt 121;54 + 1/2 n m 4 33/24 - 3/4 = 34/28 8 312% - 3/2 = 3/27 5 3°/27 - 3/4 = 3/2? = 60,57 E vo 60,57 : 1 1/27 = 63,13 Ey uy m 71 1/9 = 71:07 oa E m / 71 1/9 - 1 1/8 = 80 n mnA 80 = 91326 ga xe fore the corresponding string ratio is 1 : 4/3 = 4/3 @" LO’ = 27/3. Similarly in the third star diagram, the one for mh 91:26 ‘11/7 -11/27= 94:49 In the second of the star diagrams in Fig. 10.7, the one for the gablitu mode, the tightened string 5 T is reached from string 2 by a descending fourth. There-

iSartu| B A# A G# G F# F E# E D# D C# C B Ptolemy’s 2 3T #3 4T 4 ST 5 6T 6 7T 7 8T 8 9 the Seven tonoi in the Case 35/27| 2 1 37/23 34/26 36/2° 3/2 33/24 Construction of of the Ditonic Diatonic Genus tt tt tt 273 2 tt tt ni$tubri | » ti " " " « tt q ablitu : | nid qabli 1 tt Ptolemy does not ex- | 24/32 n plain what is the origin of tt " why the seven tónoi can be obtained by means of the 3 the Greater Perfect System, tt ” 213 " s ul, . , , a gs . , 28/33 " " " e "n " |" " " " " " „ , construction embübu " Li 10.6 Umui artul au bers 2 13 lu in Fig. 12.3 above, and why the numfor the octave-forms of the mentioned “familiar genera” do not stay strictly between the expected boundaries 60 and 120. The Fig. 12.5. Hypothetical string ratios for the seven Babylonian (ditonic) diatonic modes. common answer questions is that to these all the the nis tuhri mode, the tightened string 8 T is reached various genera appearing in ancient Greek music theory from 5 T by another descending fourth. Therefore, the are simply modifications of one basic genus, the so corresponding string ratio for string 8 T is 27/3 - 4/3 called “ditonic” diatonic, in which all intervals are = 24/32, In the same way, the string ratio is 2*/3? - 2/3 made up of tones and semitones, in particular the = 2°/3? for string 4 T, it is 2°/3? : 4/3 = 27/3* for string tetrachords of two tones (a ditone) and a semitone. 7 T, it is 27/34 - 2/3 = 28/3° for string 3 T, and it is 2*/3* This is the genus of the scale constructed in Sectio - 4/3 = 2!°/3° for string 6 T. Canonis, Prop. 20 (Fig. 12.2 above), which in its turn The result of this series of computations is disis Closely related to the seven Old Babylonian diatonic played in tabular form below. modes. There is no doubt that Ptolemy must have been In the case of the ditonic diatonic genus, an octavefamiliar with the numbers in this table, which 1s like form is a particular distribution of tones and semitones the tables in Harmonics 11.15, but in the case of the within an octave or double octave. It is instructive to ditonic diatonic genus. Incidentally, the numbers which see how the seven octave-forms can be generated in a Ptolemy associated in his tables with the dynamic surprisingly simple way in this special case. mése in each one of the seven octave-forms are the Now, return to Ptolemy’s result in Harmonics 11.10 numbers associated above with strings 3, 4 T, 5 T, 6, that the seven tónoi he had constructed (and their 7 T, 8 T, and 9. Cf. Fig. 12.8 below. corresponding octave-forms), namely A, F, D, B, G, As is well known, an important role was played in E, C, in this order, exceed each other by a semitone, two tones, Babylonian mathematics by so called “regular sexaa semitone, and two tones. Take it for gesimal numbers” defined as numbers con(ditonic) diatonic string ratios taining no other factors than positive or negative powers of 2, 3, or 5. Regular sexagesimal numbers have the important common fractions property that both they and their recipromi a) cal numbers can be expressed as integers 3 divided by suitable powers of the base 60. AT (GA) Interestingly, but purely by coincidence, 4 all the hypothetical string ratios for the ST (F#) seven Babylonian diatonic heptatonic E (A) (6) (F) modes displayed above in Fig. 12.5 are 6T (E#) regular sexagesimal numbers. In Fig. 12.6, 6 (E) all those string ratios are written first as TT (DA) ratios of powers of 2 and 3, then as 7 common fractions, and finally as sexages- 8 T (C#) imal numbers, both in the Babylonian > form and in the form used by Ptolemy in his tables. D 5) sexagesimal fractions asin Harmonics 11.15 1 1 1 60 2°/3° 3/2 213° 3/2 256/243 9/8 32/27 81/64 1:03 12 35 33 20 1,07 30 1,11 06 40 1:15 56 15 63:13 67;30 71:07 75,56 2/3 3/2? 2/3 4/3 729/512 1024/729 1;20 1;25 25 46 52 30 1:24 16 47 24 26 40 80 85:26 84:17 3/2 3/2 1:30 90 2/13 372° 23 3°/2' 128/81 27/16 16/9 243/128 1:34 48 53 20 1:4115 1:46 40 1:53 54 22 30 94:49 101:15 106;40 113;54 2 2 Fig. 12.6. (Ditonic) diatonic string ratios ordered by size.

then string 3, a fifth above > = aa PR — Pea ~ è di "2 fa ” S| | a & > | e © 6, a fourth below string . = st string 7, and finally string A: the "higher" fónos a = = | ded F: a limma (semitone) below A 3. In this whole retuning ” ~ ~ ~ . tone and a semitone below A algorithm, strings 2 and en | D:a + | B: two tones and a semitone below A ~ | G:asemitone two tones and a semitone below A 2 E : a tone, a semitone two tones and a semitone below A 9 are not changed. In PS, SS ~ = S| =“ ST a} _ pes | al € al a uN >| = “| 7 . > F D en, > <*> O E C el =| €]el ©} SIT e : two tones, a semitone, two tones and a semitone below A < =< procedure in Harmonics 11.10, AY > Ptolemy’s . on the other hand, limited to the case of the ditonic diatonic genus, the seven Greek octaveforms are constructed by Fig. 12.7. Generation of the seven octave-forms in the starting with ditonic diatonic genus. (Not in Harmonics 11.10.) lydian the Mixooctave-form and then rotating that octavegranted that the octave-forms stay strictly within the form twice by a fourth downwards, then by a boundaries of the prescribed interval. Then the ocupwards, by a fourth downwards, by a fifth upwards, fifth tave-form F, being a semitone below A, must begin and by a fourth downwards. /n this whole generating with a semitone. See Fig. algorithm, 12.7 above. Similarly, the octave-form D, being a tone below F, hence a semitone and a tone below A, must begin with a tone and the boundaries of the octave are not changed. The obvious similarity between the two algoritha semitone. And so on. The final octave-form, C, must mic begin with two tones, a semitone, two tones, and a Babylonian retuning algorithm and Ptolemy’s generatsemitone, ing algorithm must be mathematically equivalent in together four tones and two semitones. procedures immediately suggests that the Old Clearly, to complete the octave, the last interval of the some sense. A simple way of demonstrating this equivaoctave-form C must then be a tone. Now, when the lence is by use of 7/3 star diagrams as in Fig. 12.8 whole octave-form C is known, the whole octave-form below. Indeed, start with the first 7/3 star diagram, E, being a tone above C will also be known. And so which on. See again Fig. 12.7. Thus, the layout of the seven corresponding to A Mixolydian. In this configuration, octave-forms in the ditonic diatonic genus is comthe octave extending from string 2, corresponding to is in the configuration of the isartu mode, pletely determined by the restriction to a fixed octave, thetic n m = disjoined néte, to string 9, corresponding together with Ptolemy’s assumption in Harmonics 11.10 to thetic u A = middle hypate, is divided, in descendthat they are dependent on each other as in Fig. 12.3. ing order, into a tone and two tetrachords. In particu- The distribution of tones and semitones (in delar, there is a tritone between strings 2 and 5. The scending order) in the seven Greek ditonic diatonic second star diagram in Fig. 12.8 is in the configuration octave-forms (as in Fig. 12.7 above) can be compared of the gablitu mode, corresponding to B Dorian. It is with the corresponding distribution of tones and semithe result of a rotation by a fourth downwards of the tones (from string 2 to string 9) in the seven Old configuration in the first start diagram. That rotation Babylonian diatonic modes (as in Fig. moves, in particular, the semitone between strings | 10.7 above). The proposed identifications are as follows: (8) and 2 (9) to a semitone between strings 4 and 5, Mixolydian and the semitone between strings 5 and 6 to a semitttstts isartu (‘normal’) Lydian stttstt embübu Phrygian tstttst nid gabli Dorian ttsttts gablitu (‘middle’) tone between 1 (8) and 2 (9). In other words, the combined effect of the rotation is that it changes the semitone between strings 5 and 6 to a semitone be- Hypolydian sttsttt kitmu Hypophrygian tsttstt pitu tween strings 4 and 5. Therefore, the only observable Hypodorian ttsttst nis tuhri result of the rotation is that it tightens string 5, which According to the retuning algorithm in UET VII 74+, § 2, the seven Old Babylonian diatonic modes is the same as moving n m = next to mése a semitone upwards. In the same way, the only observable result can be constructed by starting with a sammü instruof moving the configuration in the second star diagram ment tuned to successively by a fourth downwards is that string 1 (8) is tightened. tightening first string 5, a fourth below string 2, then Another rotation four steps downwards leads to the the isartu mode, then strings 8, a fourth below string 5 (and the equivalent configuration of the nis tuhri mode (C = Hypodorian). string 1, a fifth above string 5), then string 4, a fifth Next, a rotation five steps upwards (to the left) leads above string 8, then string 7, a fourth below string 4, to the configuration of the nid gabli mode (D Phrygian),

A Mixolydian B Dorian C Hypodorian D Phrygian 13. Cyclic Representations isartu gablitu nis tuhri nid qabli Oyf Modes in a Medie- 1 (8) 1 (8)T 1(8)T val Islamic Manuscript! Perhaps the most influential of all medieval Islamic treatises on music was Kitab al-Adwar (The Book of Cy- E Hypophrygian G Hypolydian pitu kitmu cles) by Safi ad-Din al-Urmawi (f 1294). A study of a preliminary version of that work was published by Wright O = mése in BSOAS 58. (See also Manik, ATM and Wright, Modal Systems.) A partial French translation of the work is contained within of a Fig. 12.8. Ptolemy’s construction of the seven Greek octavethe translation commentary on it in D’Erlanger, MA 3. A beautiforms, explained in terms of 7/3 star diagrams. fully written copy of Kitab al- Adwar, manuscript ljs235 a rotation four steps downwards leads to the pitu mode in the L. J. Schoenberg Collection, is available online at (E Hypophrygian), a new rotation five steps upwards http://dewey.library.upenn.edu/sceti/ljs. leads to the embubu mode (F Lydian). After the sixth the work contains diagrams presented with an exceprotation, the final configuration is in the kitmu mode, tionally high corresponding to G Hypolydian, with all strings exclarity. degree of care, This precision copy and of visual cept string 2 (9) tightened. This means that in this whole retuning algorithm, interpreted as successive tightenings of strings, strings 2 (thetic disjoined nete) and 9 (thetic middle hypdte) are never affected. That is as it should be, because all the seven octave-forms are supposed to stay strictly within the fixed central octave of the GPS. (Alternatively, following UET VII 74+, § 3, the seven Old Babylonian diatonic modes can be constructed as in Fig. 10.5 by starting again with the sammu instrument tuned to the isartu mode, then successively loosening strings 2, 6, 3, 7, 4, 1, and 8. In this alternative retuning algorithm, only string 5, halfway between strings I and 9, is never affected.) third cycle Fig. 13.1. Kitab al-Adwar, Ch. 6. Cyclic representations of three diatonic modes. The detail from the manuscript is published here with the kind permission of L. J. Schoenberg.

Of particular interest in connection with the discusthe insides of the peripheries, the Arabic letters f and sion above of the 7/3 star diagram on CBS 1766 and b denote tones and semitones, respectively. its conjectured use in Babylonian music theory are six The so called abjad numerals are based on an older circular diagrams on p. 8 of the manuscript 1js235. form of the Arabic alphabet, which begins with the Three of those diagrams are reproduced in Fig. 13.1 letters a, b, j, d. The first 9 letters stand for the ones, above, together with line drawings showing the same from 1 to 9, the next 9 letters stand for the tens, from three circles but with English translations of the Ara- 10 to 90, and so on. In Kitab al-Adwar, bic text in and around the original diagrams. “Pythagorean” notes within an octave are denoted by Along the outside of the periphery of each one of the numbers from the three circles eight numbers in alphabetic abjad 1 to 17 in abjad notation. below. (Cf. Manik, ATM, 54-56). 1 2 (3) 4 5 (6) 7 8 9 (10) 11 12 (13) 14 15 16 (17) 18 C Db (Ebb) D Eb (Fb) E F Gb (Abb) G Ab (Bbb) A Bb B (Dbb) C' S C S 5 C S S The positions of the notes within the octave are as shown numerals denote eight notes within an octave. Along S 17 fixed S e S S Cc S S S e Here s stands for a limma or semitone (string ratio cated in the cyclic diagrams in Fig. 13.1 are an octave 256/243), while c stands for a “Pythagorean comma” apart, but in contrast to the identical representations of (string ratio 531441/524288). Note that a whole tone strings 1 and 8 in the 7/3 star figure on CBS 1766, in can be divided into two semitones and a comma. It is the diagrams in Fig. 13.1 the first and eighth notes are easy to check that the notes and numbers within separated by a gap called ‘relationship of the double,’ brackets above do not appear in the cyclic representameaning “duple ratio” (ratio of the octave). tions of the three diatonic modes in Fig. 13.1. Another difference between the diagrams in Fig. Safi ad-Din’s 17 notes were constructed by use of 10.7 and those in Fig. 13.1 is that in the latter ones the an algorithm resembling the algorithm used in the unclear dichords are not represented by sides of the Sectio Canonis, Props. 19-20 (Figs. 12.1-2 above), in star diagrams. Thus, in the first diagram in Fig. 13.1, terms of only octaves, fifths, fourths, and whole tones. the unclear dichord could have been indicated by a See Manik, ATM, Ch. 3, and Fig. 13.2 below. dashed straight line connecting the third note to the With Safi ad-Din’s notations, the eight notes and seventh note, and so on.!! seven intervals of the cyclic diagrams in Fig. 13.1 are In spite of the mentioned differences between the from left to right along the peripheries of the circles, diagrams in Fig. 13.1 and those in Fig. 10.7, it is clear in descending order: first circle: 18 second circle: e that they are basically (0 15 (5 14 (t) Ich (t) Ay 18 (t) 15 (t) C’ Bb C third circle: Bb @ 11 (0 8 (s) È (t) . 12 (0 9 Ab Gb A (0) 4 N (8) (s) 8 (0 5 (0 F Eb G (SS 7 (t) F @ 1 C ever, it would be diffihi (t) È cult to believe that there 2 (5) 1 any Ristorica) sonne Db C E C of the same type. How- 1 Babylonian star diagram This means that in each one of the three cases the in CBS 1766 and the more elaborate cyclic diagrams octave is divided, in descending direction, into a whole in Kitab al-Adwar. tone, the “upper disjunction,” followed by two con- The three remaining cyclic diagrams on p. 8 of the secutive identical ditonic diatonic tetrachords. Note Schoenberg copy of Kitab al-Adwar, called ‘fourth that in contrast to the diagrams in Fig. 10.7, where all cycle,’ ‘fifth cycle,’ and ‘sixth cycle’ are of the same the sections of the peripheries of the circles are of general type, but represent three modes of a different equal length, the sections of the peripheries of the genus. corresponding to the tones are For comparison, here are the eight notes and seven larger than the sections corresponding to the semicircles in Fig. 13.1 intervals along the peripheries of each one of these tones. Moreover, the first and last of the notes indifourth circle: 18 (t) C’ fifth circle: 18 18 C’ (c,s) Bb (t) C’ sixth circle: 15 15 15 Bb three additional cyclic diagrams: (ss) Bbb (0 Bb (t) 13 12 13 Bbb (t) G (cs) Ab (c,s) 1 10 (s,s) Abb (t) 10 Abb 8 (c,s) F 8 (t) F (s,s) 8 F 6 (s, 8) Fb 5 (cs) Eb (c,s) 6 Fb 4 (t) D 3 (8,8) Ebb (t) 3 Ebb | C 1 C (ss) | C I In BSOAS 58, 467, Wright explains the situation by writing that “the notes of the scale are marked around the interpretation given in a medieval Arabic commentary to circumference and lines are drawn across to link those which Kitab al-Adwar written by Mubarak Sah. (See D’Erlanger, are a fourth or fifth apart in order to show the number of MA Ill, 324-333.) My sincere thanks are due to O. Wright consonant intervals each scale contains.” This is also the for helping me to understand some difficulties in this text.

_ 1 ] 2 3 4 5 6 “H 2 Db È a 3 on3-Ebb un + H 5 7 9 10 + 4 11 (12) (13) (14) (15) (16) Eb ® = 17 S = H 10 A H 12 A an 14 A „an 15 Bb 2 v= | | “eS, o H 13 Bbb q ra = al SF vi n ® SH TE nt 8 E v= - | Ye - | - ps 3 È = E Y i ì - 4 16 B À 1.Divide 1-m in 2 equal parts, and set 18 at the midpoint. 2. Divide 1-m in 3 equal parts, and set 11 at the end of the first 3rd. 3. Divide 1-m in 4 equal parts, and set 8 at the end of the first 4th. 4. Divide 8-m in 4 equal parts, and set 15 at the end of the first 4th. 5. Divide 1-m in 9 equal parts, and set 4 at the end of the first 9th 6. Divide 4-m in 9 equal parts and set 7 at the end of the first 9th. 7. Set 5 at the point where 8-m is prolonged by an 8th of itself in the direction of 1. 8. Set 2 at the point where 5-m is prolonged by an 8th of itself in the direction of 1. 9. Divide 2-m in three equal parts, and set 12 at the end of the first 3rd. 10. Divide 2-m in 4 equal parts, and set 9 at the end of the first 4th. 11. Divide 9-m in 4 equal parts, and set 16 at the end of the first 4th. 12. Set 6 at the point where 16-m is prolonged by one half ofitself in the direction of 1. m And so on. Fig. 13.2. The algorithm for Safi ad-Din’s division of the octave into 17 intervals. Here c, s stands for an “apotome,” a comma followed CBS 1766” Music Theory Spectrum 30 (2008), 326-338. by a limma, with the string ratio 256/243 : 65536/59049 (Overlaps certain parts of the present paper, but arrived to = 2187/2048, while s, s stands for a “double limma,” with the string ratio 256/243 - 256/243 = 65536/59049. Both kinds of intervals are indifferently written as j in the diagrams, while the limma and the comma both are written as b. late to be taken into consideration.) Deimel, A., Die Inschriften aus Fara II. Schultexte aus Fara (WVDOG 43). Leipzig, 1923, Hinrichs. Dumbrill, R. J., The Archaeomusicology of the Ancient Near East. Victoria, BC, Canada, 2005, Trafford Publishing. Englund, R.K. and J.-P. Grégoire, MSVO 1: The Proto- Cuneiform Texts from Jemdet Nasr (MSVO 1). Berlin, 1991, Gebr. Mann Verlag. Abbreviations and Bibliography D’Erlanger, R., MA: La musique arabe, 11. Paris, 1938, P. Geuthner. Abbreviations, if not listed here, follow the housestyle of AO. Friberg, J., “‘Seed and Reeds Continued.’ Another metromathematical topic text from Late Babylonian Uruk,” BaM 28 (1997), 251-365, pl. 45-46. —, Changing Views: “Bricks and mud in metro-mathematical appr. = approximately cuneiform texts, in J. Hoyrup and P. Damerow (eds.), csm = common seed measure Changing Views on Ancient Near Eastern Mathematics. GPS = Greater Perfect System Berlin, 2001, Al-Urmawi, Safi ad-Din, Kitab al-Adwar, manuscript 1js235 Max Planck Institute for the History of Science, 61-154. in the Schoenberg Collection (http://dewey.library.upenn. -, CDLJ : “On the alleged counting with sexagesimal place edu/sceti/ljs/pagelevel/index.cfm?option=view&manID value numbers in mathematical cuneiform texts from the third millennium B.C.,” Cuneiform Digital Library Jour- =]js235). Barker, A., GMW: Greek Musical Writings. Vol. Il: Harmonic and Acoustic Theory. Cambridge, UK, 1989, Cam- Bongenaar, A. C. V.M., The Neo-Babylonian Ebabbar Tem- -, MSCT 1: A Remarkable Collection of Babylonian Mathematical Texts. (Manuscripts in the Scheven Collection: (MDP 34). Paris, 1961, P. Geuthner. and mathematical -, Amazing Traces ofa Babylonian Origin in Greek Mathematics. Singapore, 2007, World Scientific. ple at Sippar, PIHANS LXXX, 1997. Bruins, E.M. and M. Rutten, Textes mathematiques de Suse L., “A musical -, Unexpected Links between Egyptian and Babylonian Mathematics. Singapore, 2005, World Scientific. bridge University Press. Crickmore, context for Cuneiform Texts 1). New York, 2007, Springer.

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